Embedding of countable abelian group into a finitely generated group

Question

Prove/ Disprove/ or object

Let, $G=\langle p,q\rangle$ be a group. Then there exists an abelian group $A$ with $\left| A\right| \leq\left|{\mathbb{N}}\right|$ and such that for every group homomorphism $\phi :A \to G$ we have that $\phi$ is not an injection.

My thought: I hold that $\mathbb{Q}$ is an $A$ that works for arbitrary $G$. Does $(\mathbb{Q},+)$ have any injections into any finitely generated group?