Consider $m \equiv 0,2 (mod 4)$ we get that the polynomial gives $a$ and $a+2$ $(mod 4)$ respectively. We can therefore see that one of them would not be a prefect square as $x^2 \equiv 0,1 (mod 4)$ We therefore get that at least $ \lfloor \frac{n}{4} \rfloor $ values of $m \leq n$ that makes the polynomial not a perfect square.