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For geodesic metric spaces, a quasi-isometry preserves the property of being a Gromov-hyperbolic metric space. Images of metric geodesic sides are uniform quasigeodesics. The Morse lemma for quasi-geodesics keeps them uniformly close to metric geodesic sides in the hyperbolic target. Thinness there and the coarse lower distance bound pull a uniform triangle-thinness constant back to the source. Coarse surjectivity supplies a quasi-inverse for the reverse implication. Consequently a metric geodesic space quasi-isometric to a tree is hyperbolic, with no local finiteness assumption on the tree.

Choose a finite generating set $T$ of $H$. A $d_T$-geodesic between two elements of $H$ maps under the inclusion to a uniform quasigeodesic in ${\mathrm{Cay}}_S(G)$ because $H$ is quasi-isometrically embedded. Since $G$ is a hyperbolic group, the Morse lemma for quasi-geodesics gives a constant $R$ such that this quasigeodesic and the ambient geodesic with the same endpoints have Hausdorff distance at most $R$. Every vertex of the former lies in $H$, so the latter lies in the closed $R$-neighborhood of $H$. Therefore $H$ is a quasiconvex subgroup.