  The purpose of this paper is to give presentations for projective $S$-unit
groups of the Hurwitz order in Hamilton's quaternions over the rational field
$\mathbb{Q}$. To our knowledge, this provides the first explicit presentations
of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In
particular, we give presentations for groups acting irreducibly and cocompactly
on a product of Bruhat--Tits trees. We also include some discussion and
experimentation related to the congruence subgroup problem, which is open when
$S$ contains at least two odd primes. In the appendix, we provide code that
allows the reader to compute presentations for an arbitrary finite set $S$.
