Let me convert my comments into an answer.

There is no such **[EDIT:] finite** graph $G$. Indeed, something stronger can be said. Suppose that a group $\Gamma$ acts transitively by permutations on a finite set $X$ , with $\#X \geq 2$. Then there is some $\gamma \in \Gamma$ for which $\gamma\colon X \to X$ has no fixed points. The proof is a simple application of Burnside's Lemma, see e.g. https://math.stackexchange.com/questions/106158/every-transitive-permutation-group-has-a-fixed-point-free-element.