Testing low degree bivariate polynomials
If you only care about the first variable, then I believe the following simple test should work: pick a random value $\beta \in \rm{GF}(q)$ and check that $f(\cdot, \beta)$ is a univariate polynomial of degree at most $d$ - say, by choosing few collections of $d+2$ points of the form $(\cdot,\beta)$ and interpolating them.

The analysis of this test should be trivial: If the function is far from being of degree $d$ in the first variable, it must be far from a univariate polynomial of degree $\le d$ for many values of $\beta$.

If you wish to test that $f$ is of degree $\le d$ in both variables, you can use the (highly non-trivial) result of Polischuk and Spielman from 1994, which is proved in Sections 2-5 in the following paper (see Theorem 9):
http://cs-www.cs.yale.edu/homes/spielman/Research/holographic.html