By playing off traces versus Dehn-Thurston coordinates, easy constructions arise.

As suggested in the comments by @YCor, a matrix representation can be written down without much trouble when expressed in terms of Dehn-Thurston coordinates on Teichmuller space that he suggested, one length-twist coordinate pair for each pants curve, except that a pants curve which is a cusp (at least one such, in order that the quotient is not compact) will not have any coordinates corresponding to it. Converting Dehn-Thurston coordinates into matrix coordinates of a corresponding representation is a straightforward one-time operation for each topological type of surface and each pants decomposition of that surface: one obtains explicit matrices for explicit generators, where the entries of the matrix are explicit functions of the lengths and twists.

As said in another comment, each group commensurate with $SL_2(\mathbb{Z})$ can be commensurated into $SL_2(\mathbb{Q})$, and therefore the traces are rational.

One also knows that for a loxodromic isometry of translation length $L$, the trace equals $2 \sinh(L/2)$ (I think I got that right...), and a closed geodesic of length $L$ corresponds to a loxodromic isometry of translation length $L$.

So just pick a number $L$ for which $2 \sinh(L/2)$ is not rational, and use $L$ as one of the pants curve lengths in the Dehn-Thurston length coordinates.

non-compactRiemann surfaces. In practice is it possible to give an explicit example of such a group $\Gamma$, by explicit I mean: explicit generators where the entries of each matrix can be viewed as the zeros of some "reasonable function" (e.g. hyper-geometric functions, or some function which satisfies a simple differential equation) ? $\endgroup$ – Hugo Chapdelaine Jul 2 '16 at 9:46