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Added 2. Let me provide the details for the case $k=2$. Without loss of generality, $$Q(x,y)=ax^2+bxy+cy^2$$ is a reduced form. That is, $$|b|\leq a\leq c,\qquad\text{whence also}\qquad a\ll\det(Q)^{1/2}.$$ The equation $Q(x,y)=n$ can be rewritten as $$(2ax+by)^2+4\det(Q)y^2=4an.$$ We can assume that there are (integral) solutions with nonzero $y$, for otherwise there are at most two solutions. In this case, $$n\geq\det(Q)/a\gg a.$$ The equation factors in the ring of integers of an imaginary quadratic number field, hence a standard divisor bound argument combined with the previous display yields that the number of solutions is $$\ll_\epsilon(an)^{\epsilon/2}\ll_\epsilon n^\epsilon.$$

Added 2. Let me provide the details for the case $k=2$. Without loss of generality, $$Q(x,y)=ax^2+bxy+cy^2$$ is a reduced form. That is, $$|b|\leq a\leq c,\qquad\text{whence also}\qquad a\ll\det(Q)^{1/2}.$$ The equation $Q(x,y)=n$ can be rewritten as $$(2ax+by)^2+4\det(Q)y^2=4an.$$ We can assume that there are (integral) solutions with nonzero $y$, for otherwise there are at most two solutions. In this case, $$n\geq\det(Q)/a\gg a.$$ The equation factors in the ring of integers of an imaginary quadratic number field, hence a standard divisor bound argument combined with the previous display yields that the number of solutions is $$\ll_\epsilon(an)^\epsilon\ll_\epsilon n^{2\epsilon}.$$

This is a substantial revision of my original response. See also the new "Added" section for an important observation by Valentin Blomer, posted with his permission.

Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies $$r_Q(n)\ll_{k,\epsilon}n^{k/2-1+\epsilon}.$$ The implied constant depends only on $k$ and $\epsilon$, so it is independent of the actual coefficients of $Q$.

Proof. We induct on $k$, and for simplicity we do not indicate the dependence of implied constants on $k$. The case $k=2$ is classical and goes back to Gauss. So let $k\geq 3$, and assume that the statement holds with $k-1$ in place of $k$. We can assume that $$ Q(x_1,\dots,x_k)=\sum_{1\leq i,j\leq k} a_{ij}x_ix_j$$ is Minkowski reduced. In particular, $a_{ij}=a_{ji}$ and $$ 0<a_{11}\leq a_{22}\leq\dots\leq a_{kk}. $$ Then we have a decomposition $$ Q(x_1,\dots,x_k)=\sum_{i=1}^k h_i\left(\sum_{i\leq j\leq k}c_{ij}x_j\right)^2,$$ where $h_i\asymp a_{ii}$, $c_{ii}=1$ and $c_{ij}\ll 1$ (see Theorem 3.1 and Lemma 1.3 in Chapter 12 of Cassels: Rational quadratic forms). In particular, the coefficients of $Q$ satisfy $$ a_{11}\dots a_{kk}\asymp h_1\dots h_k=\det(Q),$$ hence also $a_{ij}\ll a_{kk}\ll\det(Q)$ and $h_k\asymp a_{kk}\gg\det(Q)^{1/k}$.

Added. I have been in touch with Valentin Blomer about the original question, and my answer above incorporated a key input from him. More importantly, he realized that the above argument together with some automorphic input allows one to prove, for the case of $k=4$ variables, the striking uniform upper bound (with an absolute implied constant) $$r_Q(n) \ll \sigma(n).$$ Here is the argument of Valentin Blomer, posted with his permission. For $n\leq\det(Q)^{10}$, the last line of the inductive proof above gives $$ r_Q(n)\ll_{\epsilon} n^{1/2+\epsilon} + n^{1+\epsilon}\det(Q)^{-1/8+\epsilon} \ll n^{79/80+2\epsilon},$$ so we can (and we shall) assume that $n>\det(Q)^{10}$. We decompose the $\theta$-series of $Q$ uniquely as $$\theta_Q(z) = E(z) +S(z) = \sum_{n=1}^\infty a(n) e(nz) + \sum_{n=1}^\infty b(n) e(nz)$$ into an Eisenstein series and a cusp form of weight $2$ and level $N$, which is the level of $Q$. Accordingly, $r_Q(n)=a(n)+b(n)$, so it suffices to show that $a(n)\ll\sigma(n)$ and $b(n)\ll\sigma(n)$. The first bound was proved by Gogishvili (Georgian Math. J. 13 (2006), 687-691.), as follows from (2) and (13)-(14) in his paper. Therefore, it suffices to prove the second bound. We write $$S = \sum_{f \in B} c(f) f$$ in terms of an orthonormal Hecke eigenbasis $B$ for $S_2(N, \chi)$, where $\chi$ is a quadratic character and the inner product is given by $$(f, g) = \int_{\Gamma_0(N)\backslash \mathcal{H}} f(z)\bar{g}(z) \frac {dx\, dy}{y^2}.$$ We write $f(z) = \sum_n \lambda_f(n) e(nz)$, so that $b(n) = \sum_f c(f) \lambda_f(n)$. We avoid any use of Eichler/Deligne, among other things because it would require us to deal with oldforms carefully. Instead, we use the Petersson formula and Weil's bound for Kloosterman sums (together with Cauchy-Schwarz and Parseval): $$\begin{split} |b(n) |^2 \| S \|_2^{-2} n^{-1} & \leq n^{-1} \sum_f |\lambda_f(n)|^2 \ll 1 + \sum_{c} \frac{1}{c} S_{\chi}(n, n, c) J_1\left(\frac{4\pi n}{c}\right)\\ & \ll 1 + \sum_{c} \frac{(n, c)^{1/2}\tau(c)}{c^{1/2}} \min\left(\frac{n}{c}, \frac{c^{1/2}}{n^{1/2}}\right) \ll_\epsilon n^{1/2 + \epsilon}, \end{split}$$ so that $$b(n) \ll_\epsilon \| S \|_2 n^{3/4 + \epsilon}.$$ We have, by Lemma 4.2 of Blomer (Acta Arith. 114 (2004), 1-21.), $$\| S \|_2 \ll_\epsilon \det(Q)^{2+\epsilon},$$ whence in the end $$b(n) \ll_\epsilon \det(Q)^{2+\epsilon} n^{3/4 + \epsilon} \leq n^{19/20+2\epsilon}.$$ This concludes the proof. We note that for the twisted Kloosterman sum, the Weil-Estermann bound is not always true for higher prime powers, see Section 9 of Knightly-Li (Mem. Amer. Math. Soc. 224 (2013), no. 1055), but it is true for the case of quadratic characters that we are using here.

Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies $$r_Q(n)\ll_{k,\epsilon}n^{k/2-1+\epsilon}.$$ The implied constant depends only on $k$ and $\epsilon$, so it is independent of the actual coefficients of $Q$.

Remark. The "Added 1" section, posted with the permission of Valentin Blomer, contains a more precise result for $k=4$.

Proof. We induct on $k$, and for simplicity we do not indicate the dependence of implied constants on $k$. The case $k=2$ is classical and goes back to Gauss (see the "Added 2" section for more details). So let $k\geq 3$, and assume that the statement holds with $k-1$ in place of $k$. We can assume that $$ Q(x_1,\dots,x_k)=\sum_{1\leq i,j\leq k} a_{ij}x_ix_j$$ is Minkowski reduced. In particular, $a_{ij}=a_{ji}$ and $$ 0<a_{11}\leq a_{22}\leq\dots\leq a_{kk}. $$ Then we have a decomposition $$ Q(x_1,\dots,x_k)=\sum_{i=1}^k h_i\left(\sum_{i\leq j\leq k}c_{ij}x_j\right)^2,$$ where $h_i\asymp a_{ii}$, $c_{ii}=1$ and $c_{ij}\ll 1$ (see Theorem 3.1 and Lemma 1.3 in Chapter 12 of Cassels: Rational quadratic forms). In particular, the coefficients of $Q$ satisfy $$ a_{11}\dots a_{kk}\asymp h_1\dots h_k=\det(Q),$$ hence also $a_{ij}\ll a_{kk}\ll\det(Q)$ and $h_k\asymp a_{kk}\gg\det(Q)^{1/k}$.

Added 1. I have been in touch with Valentin Blomer about the original question, and my answer above incorporated a key input from him. More importantly, he realized that the above argument together with some automorphic input allows one to prove, for the case of $k=4$ variables, the striking uniform upper bound (with an absolute implied constant) $$r_Q(n) \ll \sigma(n).$$ Here is the argument of Valentin Blomer, posted with his permission. For $n\leq\det(Q)^{10}$, the last line of the inductive proof above gives $$ r_Q(n)\ll_{\epsilon} n^{1/2+\epsilon} + n^{1+\epsilon}\det(Q)^{-1/8+\epsilon} \ll n^{79/80+2\epsilon},$$ so we can (and we shall) assume that $n>\det(Q)^{10}$. We decompose the $\theta$-series of $Q$ uniquely as $$\theta_Q(z) = E(z) +S(z) = \sum_{n=1}^\infty a(n) e(nz) + \sum_{n=1}^\infty b(n) e(nz)$$ into an Eisenstein series and a cusp form of weight $2$ and level $N$, which is the level of $Q$. Accordingly, $r_Q(n)=a(n)+b(n)$, so it suffices to show that $a(n)\ll\sigma(n)$ and $b(n)\ll\sigma(n)$. The first bound was proved by Gogishvili (Georgian Math. J. 13 (2006), 687-691.), as follows from (2) and (13)-(14) in his paper. Therefore, it suffices to prove the second bound. We write $$S = \sum_{f \in B} c(f) f$$ in terms of an orthonormal Hecke eigenbasis $B$ for $S_2(N, \chi)$, where $\chi$ is a quadratic character and the inner product is given by $$(f, g) = \int_{\Gamma_0(N)\backslash \mathcal{H}} f(z)\bar{g}(z) \frac {dx\, dy}{y^2}.$$ We write $f(z) = \sum_n \lambda_f(n) e(nz)$, so that $b(n) = \sum_f c(f) \lambda_f(n)$. We avoid any use of Eichler/Deligne, among other things because it would require us to deal with oldforms carefully. Instead, we use the Petersson formula and Weil's bound for Kloosterman sums (together with Cauchy-Schwarz and Parseval): $$\begin{split} |b(n) |^2 \| S \|_2^{-2} n^{-1} & \leq n^{-1} \sum_f |\lambda_f(n)|^2 \ll 1 + \sum_{c} \frac{1}{c} S_{\chi}(n, n, c) J_1\left(\frac{4\pi n}{c}\right)\\ & \ll 1 + \sum_{c} \frac{(n, c)^{1/2}\tau(c)}{c^{1/2}} \min\left(\frac{n}{c}, \frac{c^{1/2}}{n^{1/2}}\right) \ll_\epsilon n^{1/2 + \epsilon}, \end{split}$$ so that $$b(n) \ll_\epsilon \| S \|_2 n^{3/4 + \epsilon}.$$ We have, by Lemma 4.2 of Blomer (Acta Arith. 114 (2004), 1-21.), $$\| S \|_2 \ll_\epsilon \det(Q)^{2+\epsilon},$$ whence in the end $$b(n) \ll_\epsilon \det(Q)^{2+\epsilon} n^{3/4 + \epsilon} \leq n^{19/20+2\epsilon}.$$ This concludes the proof. We note that for the twisted Kloosterman sum, the Weil-Estermann bound is not always true for higher prime powers, see Section 9 of Knightly-Li (Mem. Amer. Math. Soc. 224 (2013), no. 1055), but it is true for the case of quadratic characters that we are using here.

Added 2. Let me provide the details for the case $k=2$. Without loss of generality, $$Q(x,y)=ax^2+bxy+cy^2$$ is a reduced form. That is, $$|b|\leq a\leq c,\qquad\text{whence also}\qquad a\ll\det(Q)^{1/2}.$$ The equation $Q(x,y)=n$ can be rewritten as $$(2ax+by)^2+4\det(Q)y^2=4an.$$ We can assume that there are (integral) solutions with nonzero $y$, for otherwise there are at most two solutions. In this case, $$n\geq\det(Q)/a\gg a.$$ The equation factors in the ring of integers of an imaginary quadratic number field, hence a standard divisor bound argument combined with the previous display yields that the number of solutions is $$\ll_\epsilon(an)^{\epsilon/2}\ll_\epsilon n^\epsilon.$$

I am not sure that $r_Q(n)\ll n\log n$ is true (with an absolute implied constant), but I do see $r_Q(n)\ll_\epsilon n^{1+\epsilon}$. Moreover, $n^\epsilon$ could be specified more precisely in terms of the divisor function. In general, we have the following

Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies $$r_Q(n)\ll_{k,\epsilon}n^{k/2-1+\epsilon}.$$ The implied constant depends only on $k$ and $\epsilon$, so it is independent of the actual coefficients of $Q$.

Added. I have been in touch with Valentin Blomer about the original question, and my answer above incorporated a key input from him. More importantly, he realized that the above argument together with some automorphic input allows one to prove, for the case of $k=4$ variables, the striking uniform upper bound (with an absolute implied constant) $$r_Q(n) \ll \sigma(n).$$ Here is the argument of Valentin Blomer, posted with his permission. For $n\leq\det(Q)^{10}$, the last line of the inductive proof above gives $$ r_Q(n)\ll_{\epsilon} n^{1/2+\epsilon} + n^{1+\epsilon}\det(Q)^{-1/8+\epsilon} \ll n^{79/80+2\epsilon},$$ so we can (and we shall) assume that $n>\det(Q)^{10}$. We decompose the $\theta$-series of $Q$ uniquely as $$\theta_Q(z) = E(z) +S(z) = \sum_{n=1}^\infty a(n) e(nz) + \sum_{n=1}^\infty b(n) e(nz)$$ into an Eisenstein series and a cusp form of weight $2$ and level $N$, which is the level of $Q$. Accordingly, $r_Q(n)=a(n)+b(n)$, so it suffices to show that $a(n)\ll\sigma(n)$ and $b(n)\ll\sigma(n)$. The first bound was proved by Gogishvili (Georgian Math. J. 13 (2006), 687-691.), as follows from (2) and (13)-(14) in his paper. Therefore, it suffices to prove the second bound. We write $$S = \sum_{f \in B} c(f) f$$ in terms of an orthonormal Hecke eigenbasis $B$ for $S_2(N, \chi)$, where $\chi$ is a quadratic character and the inner product is given by $$(f, g) = \int_{\Gamma_0(N)\backslash \mathcal{H}} f(z)\bar{g}(z) \frac {dx\, dy}{y^2}.$$ We write $f(z) = \sum_n \lambda_f(n) e(nz)$, so that $b(n) = \sum_f c(f) \lambda_f(n)$. We avoid any use of Eichler/Deligne, among other things because it would require us to deal with oldforms carefully. Instead, we use the Petersson formula and Weil's bound for Kloosterman sums (together with Cauchy-Schwarz and Parseval): $$\begin{split} |b(n) |^2 \| S \|_2^{-2} n^{-1} & \leq n^{-1} \sum_f |\lambda_f(n)|^2 \ll 1 + \sum_{c} \frac{1}{c} S_{\chi}(n, n, c) J_1\left(\frac{4\pi n}{c}\right)\\ & \ll 1 + \sum_{c} \frac{(n, c)^{1/2}\tau(c)}{c^{1/2}} \min\left(\frac{n}{c}, \frac{c^{1/2}}{n^{1/2}}\right) \ll_\epsilon n^{1/2 + \epsilon}, \end{split}$$ so that $$b(n) \ll_\epsilon \| S \|_2 n^{3/4 + \epsilon}.$$ We have, by Lemma 4.2 of Blomer (Acta Arith. 114 (2004), 1-21.), $$\| S \|_2 \ll_\epsilon \det(Q)^{2+\epsilon},$$ whence in the end $$b(n) \ll_\epsilon \det(Q)^{2+\epsilon} n^{3/4 + \epsilon} \leq n^{19/20+2\epsilon}.$$ This concludes the proof. We note that for the twisted Kloosterman sum, the Weil-Estermann bound is not always true for higher prime powers, see Section 9 of Knightly-Li ( Mem. Amer. Math. Soc. 224 (2013), no. 1055), but it is true for the case of quadratic characters that we are using here.

Theorem. Let $Q(x_1,\dots,x_k)$ be a positive definite integral quadratic form in $k\geq 2$ variables. Then the number of integral representations $Q(x_1,\dots,x_k)=n$ satisfies $$r_Q(n)\ll_{k,\epsilon}n^{k/2-1+\epsilon}.$$ The implied constant depends only on $k$ and $\epsilon$, so it is independent of the actual coefficients of $Q$.

Added. I have been in touch with Valentin Blomer about the original question, and my answer above incorporated a key input from him. More importantly, he realized that the above argument together with some automorphic input allows one to prove, for the case of $k=4$ variables, the striking uniform upper bound (with an absolute implied constant) $$r_Q(n) \ll \sigma(n).$$ Here is the argument of Valentin Blomer, posted with his permission. For $n\leq\det(Q)^{10}$, the last line of the inductive proof above gives $$ r_Q(n)\ll_{\epsilon} n^{1/2+\epsilon} + n^{1+\epsilon}\det(Q)^{-1/8+\epsilon} \ll n^{79/80+2\epsilon},$$ so we can (and we shall) assume that $n>\det(Q)^{10}$. We decompose the $\theta$-series of $Q$ uniquely as $$\theta_Q(z) = E(z) +S(z) = \sum_{n=1}^\infty a(n) e(nz) + \sum_{n=1}^\infty b(n) e(nz)$$ into an Eisenstein series and a cusp form of weight $2$ and level $N$, which is the level of $Q$. Accordingly, $r_Q(n)=a(n)+b(n)$, so it suffices to show that $a(n)\ll\sigma(n)$ and $b(n)\ll\sigma(n)$. The first bound was proved by Gogishvili (Georgian Math. J. 13 (2006), 687-691.), as follows from (2) and (13)-(14) in his paper. Therefore, it suffices to prove the second bound. We write $$S = \sum_{f \in B} c(f) f$$ in terms of an orthonormal Hecke eigenbasis $B$ for $S_2(N, \chi)$, where $\chi$ is a quadratic character and the inner product is given by $$(f, g) = \int_{\Gamma_0(N)\backslash \mathcal{H}} f(z)\bar{g}(z) \frac {dx\, dy}{y^2}.$$ We write $f(z) = \sum_n \lambda_f(n) e(nz)$, so that $b(n) = \sum_f c(f) \lambda_f(n)$. We avoid any use of Eichler/Deligne, among other things because it would require us to deal with oldforms carefully. Instead, we use the Petersson formula and Weil's bound for Kloosterman sums (together with Cauchy-Schwarz and Parseval): $$\begin{split} |b(n) |^2 \| S \|_2^{-2} n^{-1} & \leq n^{-1} \sum_f |\lambda_f(n)|^2 \ll 1 + \sum_{c} \frac{1}{c} S_{\chi}(n, n, c) J_1\left(\frac{4\pi n}{c}\right)\\ & \ll 1 + \sum_{c} \frac{(n, c)^{1/2}\tau(c)}{c^{1/2}} \min\left(\frac{n}{c}, \frac{c^{1/2}}{n^{1/2}}\right) \ll_\epsilon n^{1/2 + \epsilon}, \end{split}$$ so that $$b(n) \ll_\epsilon \| S \|_2 n^{3/4 + \epsilon}.$$ We have, by Lemma 4.2 of Blomer (Acta Arith. 114 (2004), 1-21.), $$\| S \|_2 \ll_\epsilon \det(Q)^{2+\epsilon},$$ whence in the end $$b(n) \ll_\epsilon \det(Q)^{2+\epsilon} n^{3/4 + \epsilon} \leq n^{19/20+2\epsilon}.$$ This concludes the proof. We note that for the twisted Kloosterman sum, the Weil-Estermann bound is not always true for higher prime powers, see Section 9 of Knightly-Li (Mem. Amer. Math. Soc. 224 (2013), no. 1055), but it is true for the case of quadratic characters that we are using here.