Can the quotient of a nonzero ring be a zero ring?

Sure. For instance, let $R=2\mathbb{Z}$ and $I=4\mathbb{Z}$.

More generally, to say that $R/I$ is a zero ring just means that $R^2\subseteq I$. So, you're just asking for an example of $R$ without zero divisors such that $R^2$ is a proper ideal.