Are countable dense subspaces of $\mathbb{R}^n$ homeomorphic to ${\mathbb Q}^n$?

According to https://arxiv.org/abs/1210.1008 Example 2(c)... yes, they are all homeomorphic to $\mathbb Q$!

You can learn a bit more about countable dense subsets of separable metric spaces by searching for "countable dense homogeneous space".

A separable metric space $X$ is Countable Dense Homogeneous (CDH) if given any two countable dense subsets $D$ and $E$ of $X$ there is a homeomorphism $f : X \rightarrow X$ such that $f(D) = E$.

The concept was introduced by R. Bennett in Countable dense homogeneous spaces Fund. Math., 74 (1972), pp. 189-194

Theorem 3 in the paper implies that locally euclidean spaces are CDH. So not only any countable dense subset of ${\bf R}^n$ is homeomorphic to ${\bf Q}^n$, but the homeomorphism can be taken as the restriction of a global homeomorphism of ${\bf R}^n$.