  We initiate a new, computational approach to a classical problem: certifying
non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of
$\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski
dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$
for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that
a M\"{o}bius subgroup $G$ is not free by showing that it has finite index in
the relevant $\mathrm{SL}(2, R)$. Further information about the structure of
$G$ is obtained; for example, we compute the minimal subgroup of finite index
in $\mathrm{SL}(2,R)$ that contains $G$.
