As Ofir Gorodetsky notes in the comments, Weil's bond 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 $n$ $o(\sqrt{q})$, 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 cancelation. See: J. Bourgain's paper "[Mordel's Exponential Sum Estimate Revisted][1]"


  [1]: https://www.ams.org/journals/jams/2005-18-02/S0894-0347-05-00476-5/S0894-0347-05-00476-5.pdf