Yes, the geodesic flow on the hyperbolic plane and, in fact, on any Hadamard manifold (${\mathbb R}^n$ provided with a Riemannian metric with non-positive curvature) is integrable.

You can easily construct integrals of motion for the geodesic flow in hyperbolic space as follows: 

**1.** Consider the Cayley-Klein model where the hyperbolic space is the interior of the unit ball in ${\mathbb R}^n$ and (oriented) geodesics are (oriented) straight lines. 

**2.** The space of geodesics is then the space of oriented lines passing through the unit ball and this is diffeomorphic to the unit codisc bundle of the standard Riemannian metric on $S^{n-1}$. In particular, the space of geodesics is a smooth manifold.

**3.** Consider the canonical projection $\pi$ that associates to a unit covector 
$\xi \in T^*{\mathbb H}^n$ the oriented geodesic that has $\xi$ as initial condition.

**4.** The pull-back by $\pi$ of any smooth function on the space of geodesics (suitably extended as a homogeneous function) will be an integral of motion.

The key idea in this is that if a Riemannian or Finsler metric is such that its space of geodesics is a manifold, then the geodesic flow is integrable. This has been worked out in detail by Carlos E. Duran in 

*On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed* Ergodic Theory and Dynamical Systems (1997), 17 : pp 1359-1370 1997.

He considers the compact case, but he also works out the case where the space of geodesics is an orbifold (like lens spaces).