We explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator $(-\Delta_{\mathbb{B}^N})^{-\alpha/2}$ in the hyperbolic space \[-\Delta_{\mathbb{B}^{N}} u \, - \, \lambda u \, = \left[(-\Delta_{\mathbb{B}^{N}})^{-\frac{\alpha}{2}}|u|^p\right]|u|^{p-2}u,\] where $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator, $\lambda \leq \frac{(N-1)^2}{4}, 1 < p < 2^*_{\alpha} = \frac{N+\alpha}{N-2}, 0 < \alpha < N, N \geq 3, 2^*_\alpha$ is the critical exponent in the context of the Hardy-Littlewood-Sobolev inequality (HLS). This study is analogous to the Choquard equation in $\mathbb{R}^N$, involving non-local Riesz potential operator. We consider the functional setting within the Sobolev space $H^1(\mathbb{B}^N)$, employing advanced harmonic analysis techniques, the Helgason Fourier transform, and semigroup approach to the fractional Laplacian. The HLS inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the problem, demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.