The existence of such a pair of graphs would contradict Hedetniemi's conjecture for finite graphs. 

Suppose $\chi(G \times H) = k < \aleph_0$ then $\chi(G_0 \times H_0) \leq k$ for all finite induced subgraphs $G_0$ and $H_0$ of $G$ and $H$, respectively. If $\chi(G) > k$ then this is witnessed by a finite induced subgraph $G_0$ of $G$. Assuming Hedetniemi's conjecture, we must then have $\min(\chi(G_0),\chi(H_0)) = \chi(G_0 \times H_0) \leq k$ for every finite induced subgraph $H_0$ of $H$. Since $\chi(G_0) > k$, we must have $\chi(H_0) \leq k$. But then $H$ must be $k$-colorable since every finite induced subgraph of $H$ is $k$-colorable.

In other words, the above shows that if Hedetniemi's conjecture holds for all finite graphs, then it also holds for infinite graphs when at least one of the factors is finitely or countably colorable.