Prove that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition.

Question

How can I show that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition: $A$ is an injective object in $\Bbb Z\text{-mod}$ iff the morphism Hom$_{\Bbb Z}(\Bbb Z,A) \rightarrow$ Hom$_{\Bbb Z}(J,A); g\mapsto g \circ i$ is surjective, in which $i:J \rightarrow \Bbb Z; j \mapsto j$ (inclusion). I found the same question which is solved using Baer's criterion here, and I do note the similarity between Baer's criterion and the definition I wrote, but I can't understand how to prove the surjective part.