Let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed). My question is if although the space time is curved then can we use the WKB approximation to get the quantum energies from the momenta
$ \oint _{C} p_{a}dq_{a}=2\pi \hbar (n_ {a}+ \mu _{a})$ and
$ \oint _{C} p_{b}dq_{b}=2\pi \hbar (n_ {b}+ \mu _{b})$
For example for the hyperbolic metric $ ds^{2} = \frac{dx^{2} +dy^{2}}{y^{2}} $ with Hamiltonian $ H= -y^{2}( \partial _{x}^{2}+ \partial _{y}^{2}) $.
Yes you can. The quantum motion on any Hamiltonian which has a classical limit which is integrable limits to the Bohr-Sommerfeld rule at large quantum numbers.
This Hamiltonian is the Laplacian corresponding to the metric on the manifold. This is a widely researched problem. The WKB approximation associates discrete eigenvalues of the Laplacian to closed geodesics (with the quantization condition) and continuous spectrum to open geodesics. The example given in the question (The hyperboloid) does not have closed geodesics, thus no quantization condition exists. Please see the following physics report by R. Campouresi section 5.5 (and in general for the spectrum geodesic duality). However, if one adds an interaction to a homogeneous magnetic field ( a covariantly constant magnetic field) to the Laplacian on the hyperboloid, it is known that closed (magnetic) geodesics will be formed and as a consequence discrete spectrum will exist.
