The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n) =\pi x \cdot \frac{\eta_{a}(q)}{q^2}+ R_{q,a}(x)$$ where $\eta_{a}(q) := \{ (x,y) \in (\mathbb{Z}/q\mathbb{Z)}^2 : x^2 +y^2 \equiv a \bmod q\}$, then $$R_{q,a}(x) = O\left( x^{\frac{2}{3} + \xi} q^{-\frac{1}{2}(1+3\xi)}\gcd(a,q)^{1/2}\tau(q) \right)$$ for any $\xi \in (0,1/3)$. This is non-trivial for $q \le x^{\frac{2}{3}-\varepsilon}$. In particular, as $x$ tends to infinity, your expression is asymptotic to $\pi x$ times a constant depending on $a \bmod q$. If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that $$R_{q,a}(x) = O\left( (q^{\frac{1}{2}}+x^{\frac{1}{3}}) \gcd(a,q)^{1/2}\tau^4(q)\log^4 x \right).$$ Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.