There might be an English translation of Vigneras book. If not, at least three is [Arithmetic of Hyperbolic 3-manifolds](http://www.springer.com/gb/book/9780387983868) 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: 

$$\text{Vol}\big( SL(2,\mathbb{C})  / \rho (\mathcal{O}  ) \big)=
\frac{|\Delta_k|^{3/2}\zeta_k(2)\prod_{\mathcal{P}\big|N(A)}\big( N(\mathcal{P}-1\big)}
{(4\pi^2)^{[k:\mathbb{Q}]-2}}$$

This volume is expressed in terms of invariants of the quaternion algebra,  which are in the textbook.  For example $k/\mathbb{Q}$ is the number field which was used to define $A$, and $\zeta_k(2)$ is the Zeta function.  There's another formula for  volumes of $SL(2,\mathbb{R})   $ quotients as well.