I hate to see an easy question without a proper answer, so I'll elaborate a bit on few_reps comment which really contains all there is to say about this question; his reference to Reid and MacLachlan's book also stands for this answer.

If $\Gamma$ is any Fuchsian group derived from a quaternion algebra then there are infinitely many maximal Fuchsian lattices commensurable to $\Gamma$. In particular, since there are up to conjugation only finitely many maximal orders this implies that there are infinitely many examples of arithmetic groups not derived from a quaternion algebra in any given commensurability class.

For an explicit example of such you can take $\Gamma={\rm SL}_2({\mathbb Z})$, a prime $p$ and
$$
\Gamma_0(p) = \left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma:\: p|b\right\}.
$$
Then the normalizer $\Gamma'$ of $\Gamma_0(p)$ in ${\rm SL}_2({\mathbb R})$ is not contained in ${\rm SL}_2({\mathbb Q})$ since
$$
\begin{pmatrix} 0&-\sqrt p\\ (\sqrt p)^{-1}&0\end{pmatrix}
$$
belongs to $\Gamma'$.

conjugateto a finite index subgroup of the normalizer of (the image in $\mathrm{PSL}_2(\mathbf R)$ of an Eichler order in some quaternion algebra (see C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds). These normalizers are often larger (than their Eichler orders), so these are examples. $\endgroup$ – few_reps Oct 21 '13 at 2:01