Let $G=GL_2(\mathbb F_p)$, $M=M_2(\mathbb F_p)$, $M^0=$ matrices of trace 0 in $M$. if you consider first $H^1(G,M^0)$, then it is 0 as soon as $p>7$ by CPS (Cline, Parshall, Scott, cohomology of finite groups of Lie type, Publ. IHES 45, (1975)) Theorem 4.2. (This is a result sometimes useful in Galois deformation theory, cf. Mazur, "Deforming Galois Representations", pages 401-402).
As for $H^1(G,M)$ you get that it has dimension 1 if $p>7$ since $M=M_0 \oplus \mathbb F_p$ as $G$-modules when $p$ is odd.