Since $GL(2,Z/p^nZ)$ is an iterated extension of $C_p^{2^2}$ by $GL(2,Z/pZ)$, one could form a presentation by adding further generators and modifying the existing relations (e.g. $a^2=1$ now becomes $a^2=n$ for a suitable element of the normal subgroup) -- this is a general technique for building presentations.
Since the pattern of repeated powering for larger $n$ is very regular, it will be sufficient to spell out the rules once for $p^2$.