Code to plot the quiver in question:

Mutating this quiver at the vertex $0$ or vertex $1$ gives a quiver that is isomorphic to the original quiver. In fact, the resultant quiver is the original quiver with all its arrows reversed.

Since vertices $0$ and $1$ are the only vertices that can be mutated at, this construction is equivalent to an ice quiver of type $\widetilde{A}_1$. So this quiver produces infinitely many cluster variables.