There might be an English translation of Vigneras book. If not, at least three is Arithmetic of Hyperbolic 3-manifolds by Colin McLachlan and Alan Reid. That book has information about quaternion algebras.
I'll look into discussion of the fundamental volume of the tiling over $\mathbb{H}^3$.
In Chapter 11 they discuss Tamagawa measure. If $A$ is a quaternion algebra, and $\mathcal{O} \subseteq A $ is a maximal order then we've got a formula for the volume:
$$\text{Vol}( SL(2,\mathbb{C}) / \rho (\mathcal(O) )= \frac{|\Delta|^{3/2}\zeta_k(2)\prod_{\mathcal(P)|N(A)}\big( N(\mathcal(P)-1\big)} {(4\pi^2)^{[k:\mathbb{Q}]-2}}$$