Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order.
What would be an example of an arithmetic Fuchsian lattice that is not of finite index in an Eichler order?
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 $\Gamma'$.
