This group is a subgroup of an S-arithmetic lattice, which acts discretely on finite-valence Bass-Serre tree, hence is virtually free.
The unit quaternions $q_i$ may be interpreted as elements in $SU(2)$, treating left multiplication as unitary transformations of $\mathbb{C}^2$. The entries to the matrices lie in $\mathbb{Q}(\sqrt{5},i)$, and traces lie in $\mathbb{Q}(\sqrt{5})$. Then $tr(q_i)=-\frac12$, and one may apply Theorem 3.6.2 of MacLachlan-Reid to conclude that the quaternion algebra generated by this group over $\mathbb{Q}(\sqrt{5})$ has Hilbert symbol $(-15, -110)$. Thus, the quaternion algebra ramifies at both real places. I consulted Matt Stover at lunch, who also explained to me that it is ramified at both places sitting over the prime $11$, and no where else. So the group acts discretely on the Bass-Serre tree associated to the prime $2$ (which doesn't split in $\mathbb{Q}(\sqrt{5})$). Thus, the group is virtually free.
I'm not sure how to prove that it's freely generated by $q_i$, without explicitly computing the action on the Bass-Serre tree. One might be able to take advantage of the $S_4$ symmetry of the generators to determine this.