Triangulating surfaces quasi-isometrically

We prove that if a complete Riemannian surface $(\Sigma,d_\Sigma)$ is quasi-isometric to some bounded degree graph $G$, then $\Sigma$ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric. We study several variants of the problem, and identify the right condition making it an if and only if statement.