This is not an answer, but a long comment providing related known results.

In the particular case $Q=I_{n,1}= diag(I_n,-1)=diag(1,\dots,1,-1)$ (called the Lorentzian case), Ratcliffe and Tschantz gave in [1] an asymptotic formula for the number of $x=(x_1,\dots,x_{n+1})^t\in \mathbb Z^{n+1}$ satisfying that $x^tI_{n,1}x=k$ ($k\in\mathbb Z$) and $\|x\|^2\leq t$, as $t\to\infty$. When $k$ is negative, they applied (in a clever way) the lattice point theorem by Lax and Phillips, obtaining a formula with an error term.

In my Ph.D. thesis (nine years ago) I extended the above formulas for more general indefinite quadratic or hermitian forms $Q$ of signature $(n,1)$ (see [2]). Furthermore, I also worked on counting certain solutions of $X\in \mathbb Z^{(p+q)\times q}$ satisfying $I_{p,q}[X]=X^*I_{p,q}X=-L$, where $L$ is a positive definite integral matrix by using a lattice point theorem by Gorodnik and Nevo [3] or [4]. Unfortunately, concerning the last case, it is not easy to compute explicitly the constants involved in the asymptotic formulas.

I hope similar technics as above may help to find upper bounds like those requested in the question.

[1] Ratcliffe, John G.; Tschantz, Steven T., On the representation of integers by the Lorentzian quadratic form, J. Funct. Anal. 150, No. 2, 498-525 (1997). ZBL0883.11017.

[2] Lauret, Emilio A., An asymptotic formula for representations of integers by indefinite Hermitian forms, Proc. Am. Math. Soc. 142, No. 1, 1-14 (2014). ZBL1329.11029.

[3] Gorodnik, Alexander; Nevo, Amos, The ergodic theory of lattice subgroups, Annals of Mathematics Studies 172. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14185-5/pbk; 978-0-691-14184-8/hbk). 136 p. (2009). ZBL1186.37004.

[4] Gorodnik, Alexander; Nevo, Amos, Counting lattice points, J. Reine Angew. Math. 663, 127-176 (2012). ZBL1248.37011.

This is not an answer, but a long comment providing related known results.

In the particular case $Q=I_{n,1}= diag(I_n,-1)=diag(1,\dots,1,-1)$ (called the Lorentzian case), Ratcliffe and Tschantz gave in 1 an asymptotic formula for the number of $x=(x_1,\dots,x_{n+1})^t\in \mathbb Z^{n+1}$ satisfying that $x^tI_{n,1}x=k$ ($k\in\mathbb Z$) and $\|x\|^2\leq t$, as $t\to\infty$. When $k$ is negative, they applied (in a clever way) the lattice point theorem by Lax and Phillips, obtaining a formula with an error term.

In my Ph.D. thesis (available at my web page, though it is written in Spanish) I extended the above formulas for more general indefinite quadratic or hermitian forms $Q$ of signature $(n,1)$ (see [2]). Furthermore, I also worked on counting certain solutions of $X\in \mathbb Z^{(p+q)\times q}$ satisfying $I_{p,q}[X]=X^*I_{p,q}X=-L$, where $L$ is a positive definite integral matrix by using a lattice point theorem by Gorodnik and Nevo [3] or [4]. Unfortunately, concerning the last case, it is not easy to compute explicitly the constants involved in the asymptotic formulas.

I hope similar technics as above may help to find upper bounds like those requested in the question.

1 Ratcliffe, John G.; Tschantz, Steven T., On the representation of integers by the Lorentzian quadratic form, J. Funct. Anal. 150, No. 2, 498-525 (1997). ZBL0883.11017.

[2] Lauret, Emilio A., An asymptotic formula for representations of integers by indefinite Hermitian forms, Proc. Am. Math. Soc. 142, No. 1, 1-14 (2014). ZBL1329.11029.

[3] Gorodnik, Alexander; Nevo, Amos, The ergodic theory of lattice subgroups, Annals of Mathematics Studies 172. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14185-5/pbk; 978-0-691-14184-8/hbk). 136 p. (2009). ZBL1186.37004.

[4] Gorodnik, Alexander; Nevo, Amos, Counting lattice points, J. Reine Angew. Math. 663, 127-176 (2012). ZBL1248.37011.