Is $\mathbb{Z \times Q }$ with lexicographical order isomorphic to $\mathbb{Z \times Z }$ with lexicographical order?

I think that $\{\mathbb{Z \times Q}, \leq_{lex}\}$ is not isomorphic to $\{\mathbb{Z \times Z }, \leq_{lex}\}$, but I am not sure if my argumentation is sufficient. Is it enough to state that $\{\mathbb{Z \times Q}, \leq_{lex}\}$ is a dense order, while $\{\mathbb{Z \times Z }, \leq_{lex}\}$ is not?