Let R = $Mat_2(F)$ be the ring of all 2 by 2 matrices over a finite field F, X the set of all nonzero, nonunits of R and G the group of all units of R. After investigating some properties of orbits under the left (and right) regular action on X by G, we show that the graph automorphisms group of $Gamma(R)$ (the zero-divisor graph of R) is isomorphic to the symmetric group $S_{|F|+1}$ of degree |F|+1.