As Ofir Gorodetsky notes in the comments, Weil's bound gives that the absolute value of the sum is most order $\max(m,n) \sqrt{q}$. This is non-trivial as long as $m$ and $n$ are $o(\sqrt{q})$, after this the estimate is worse than the trivial bound.
When $m$ and $n$ are large there are known estimates from additive combinatorics, at least when $q$ is prime. There are some obstructions to estimates, particularly when $m$, $n$ or $m-n$ have a large common factor with $q-1$. Otherwise one can obtain cancellation. See: J. Bourgain's paper "Mordell's Exponential Sum Estimate Revisited".
