  Let $x$, $y$ be two integral quaternions of norm $p$ and $l$, respectively,
where $p$, $l$ are distinct odd prime numbers. We investigate the structure of
$<x,y>$, the multiplicative group generated by $x$ and $y$. Under a certain
condition which excludes $<x,y>$ from being free or abelian, we show for
example that $<x,y>$, its center, commutator subgroup and abelianization are
finitely presented infinite groups. We give many examples where our condition
is satisfied and compute as an illustration a finite presentation of the group
$<1+j+k, 1+2j>$ having these two generators and seven relations. In a second
part, we study the basic question whether there exist commuting quaternions $x$
and $y$ for fixed $p$, $l$, using results on prime numbers of the form $r^2 + m
s^2$ and a simple invariant for commutativity.
