I thought this problem is really easy and give the following 5 line proof:
We assume $R$ is commutative as the book assumed. Suppose we have a non-zero homomorphism $\phi$, then $\ker(\phi)$ is a proper ideal of $Q$. But $Q$ is a field and has only two ideals. Since we assume $\ker(\phi)\not=Q$, $\phi$ must be injective. But this is impossible, since $\phi\in Hom(Q,R)$ and $1$ must be mapped to 1, every non-zero element in $r$ now has an inverse from $\phi(1/r)$. Since $R$ is not a field this is absurd.
However the professor give me zero mark on this problem. I think I might have ignored something really, really basic. So I venture to ask at here. I apologize for my shallowness. I read this again, and other than $R$ could be non-commutative I do not see where it went wrong.