$M$ is a flat $R$-module if and only if $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is injective.
One direction is easy. Suppose $M$ is flat. We know that
$$ \hom_\mathbb{Z}(-\otimes M, \mathbb{Q/Z}) \cong_{\mathbb{Z}} \hom_{\mathbb{Z}}(-,\hom_{\mathbb{Z}}(M,\mathbb{Q/Z}))$$
Since $- \otimes M$ is exact and $\mathbb{Q/Z}$ is injective, the left functor is exact which shows that the right functor is exact, i.e. $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is injective.
The other direction looks difficult and indeed I haven't been able to prove it attacking the problem by considering short exact sequences. The thing is I don't see why $\mathbb{Q/Z}$ is special in this problem and I think I need to figure that out to be able to come up with a proof. Any help is appreciated.