 Solution

For a finite generating set $S$ of $G$, the Cayley graph ${\mathrm{Cay}}_S(G)$ has vertex set $G$ and an edge from $g$ to $gs$ for every $g\in G$ and $s\in S\cup S^{-1}$; one may retain orientations and labels, or forget them.

As unlabelled undirected graphs, the Cayley graph of the infinite cyclic group $\mathbb{Z}$ with generator $1$ and the Cayley graph of the infinite dihedral groupwith generators $a,b$ are both the two-way infinite line. The first group is an abelian group and the second is not, so they are not isomorphic. Thus the requested example is the infinite line as a Cayley graph of $\mathbb{Z}$ and $D_{\infty }$.