When $R$ is noetherian, yes:
 
By Baer's criterion it suffices to prove that $\hom_{S^{-1} R}(S^{-1} R,S^{-1} M) \to \hom_{S^{-1} R}(S^{-1} I,S^{-1} M)$ is surjective for every ideal $I \subseteq R$. Since $I$ (and of course $R$) are of finite presentation, this map is isomorphic to $S^{-1} \hom_R(R,M) \to S^{-1} \hom_R(I,M)$, which is surjective since $\hom_R(R,M) \to \hom_R(I,M)$ is surjective.