I have been trying to find a presentation of the upper triangular subgroup of $GL(2, \mathbb Q)$ by considering the free group $Fr(\{x_i| i\in \mathbb Q\})$ under a homomorphism $f$ into $GL(2, \mathbb Q)$ by
$f(x_i):=\begin{bmatrix}i&b\\&{1\over i}\end{bmatrix}$ or $f(x_i):=\begin{bmatrix}i&b\\& {i^2} \end{bmatrix}$ but I haven't been able to finagle it enough to get it to work. Is this at all the right approach? How exactly do I go about finding a presentation in the proper way? Thanks.