Let $(X,d)$ be a Gromov hyperbolic metric space, and let $\partial X$ be the Gromov boundary of $X$. Fix a group $G\leq\operatorname{Isom}(X)$ and a point $\xi\in\partial X$. We consider the Diophantine approximation of a point $\eta\in\partial X$ by points in the set $G(\xi)$. Our results generalize the work of many authors, in particular Patterson ('76) who proved most of our results in the case that $G$ is a geometrically finite Fuchsian group of the first kind and $\xi$ is a parabolic fixed point of $G$.