If $H$ is definite, then the group of units of $H$ is finite. If $H$ is infinite, then the group of units is a pretty chunky group; it embeds as a cocompact discrete subgroup of $SL(2, R)$, and the rank of its abelianization (which, if I remember correctly, can be interpreted geometrically as the genus of an associated modular curve) can be arbitrarily large.
All of this follows from general theorems on arithmetic subgroups of algebraic groups; this is a classical theory going back to Borel, Harish-Chandra and Ono in the 60s, and there is a nice summary in Gross's paper "Algebraic modular forms".