We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_λ$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree with geometric offspring distribution. The tree of infinite geodesics in $\mathbb{T}_λ$ provides a new notion of boundary, which is a realization of the Poisson boundary. By scaling limits arguments, we also obtain a description of the tree of infinite geodesics in the hyperbolic Brownian plane. Finally, by combining our main result with a forthcoming paper, we obtain new hyperbolicity properties of $\mathbb{T}_λ$: it satisfies a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.