Can we embed a countable compact subset of $\Bbb{Q}$ into the Cantor space? (Both use the standard topology.)

Question

Let X be a compact subspace of $\Bbb{Q}$ with cardinality countable, denote the Cantor space as $\mathcal{C}$. Is there a subspace $Y \subset \mathcal{C}$ and a mapping $f$ such that $f: X \rightarrow Y$ is a homeomorphism?

A further question is, can we achieve the embedding into the rationals in the Cantor space? (That is, elements of the Y mentioned above are all rationals.)