Just to make sure that it's not hidden in the comments for future readers, I want to point out that the paper "On the Approximation of Irrational Numbers With Rationals Restricted By Congruence Relations" actually gives a positive answer to a slight extension of my question (where rather than just having $\left|\frac mn-\alpha\right|\lt \frac1{n^2}$ one has that the difference is bounded for $\frac{C}{n^2}$ for some $C$ possibly dependent on the congruence parameters), which was actually sufficient to show what I was originally after. The precise statement is:

For any irrational $\xi$, any $s\geq 1$, and integers $a, b$, there are infinitely many integers $m, n$ satisfying $\displaystyle\left|\xi-\frac{m}{n}\right|\lt\frac{2s^2}{n^2}$, $m\equiv a\pmod s$, $n\equiv b\pmod s$.