You might want to study the work of Detinko, Flannery, and co-authors. For example:
Detinko, A.; Flannery, D. L.; Hulpke, A., Zariski density and computing in arithmetic groups, ZBL06825254.
Detinko, A. S.; Flannery, D. L.; Hulpke, A., Algorithms for arithmetic groups with the congruence subgroup property., J. Algebra 421, 234-259 (2015). ZBL1319.20040.
In the real hyperbolic case, there is the very interesting paper of Mark and Paupert:
Presentations for cusped arithmetic hyperbolic lattices, by Alice Mark and Julien Paupert.