Let $R$ be a ring. I want to prove that every quotient of a cyclic $R$-module is cyclic.
Can I assume that the question is about the quotient module or it could be just quotient?
Because in the first case if $M$ is a cyclic $R$-module then $M= \langle m\rangle$ for some $m$, and a quotient $N$ of $M$ would be given by a surjective $R$-module homomorphism $H:M\to N$, but this means that $N=\langle H(m)\rangle$, which is to say $N$ is cyclic.
Is this proof right?
Yes the proof is correct. (If unsure, you should explain in detail why $N = \langle H(m) \rangle$.)
If $M$ is a module, then a quotient of $M$ is a module $Q$ with a surjective homomorphism $M \to Q$. If $U$ is the kernel, it follows $Q \cong M/U$. Conversely, $M/U$ is always a quotient of $M$. So up to isomorphism there is no difference.