  We study the structure of infinite geodesics in the Planar Stochastic
Hyperbolic Triangulations $\mathbb{T}_{\lambda}$, 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 $\mathbb{T}_{\lambda}$ 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 $\mathbb{T}_{\lambda}$: it satisfies
a weaker form of Gromov-hyperbolicity and admits bi-infinite geodesics.
