Is $\mathbb{Q}$ a vector space over $\mathbb{Z}$ or $\mathbb{Q}$?

You can't have a vector space over $\Bbb Z$. By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, $\Bbb Q^n$ is definitely a $\Bbb Z$-module.

That being said, there is a list of $10$ requirements (or thereabouts, it varies slightly) for a space like $\Bbb Q^n$ to be a vector space over a field like $\Bbb Q$. All of them should be easily verifiable. So yes, $\Bbb Q^n$ is a vector space over $\Bbb Q$.