If $f:X \to Y$ and $g:Y \to X$ are continuous functions, it is well known that:
If both of them are quotient maps, then the composition $g \circ f$ is a quotient map; and
If $g \circ f$ is a quotient map, then $g$ is a quotient map.
However, there is no need of $f$ being a quotient map if we take $g \circ f$ to be a quotient map (even if we assume that $f$ is surjective); this is remarked (without the presentation of an example, and regarded as a fact of easy verification) at Engelking's book.
Indeed, after some reasoning I came out with a whole class of examples: for any $f:X \to Y$ which is surjective and continuous but does not coinduce the topology of $Y$, let $Z$ be a singleton and $g:Y \to Z$ the only possible function in this context - and here you go, the composition is a quotient map but $f$ is not.
I was thinking, are there more nicer examples ? In particular, I would like to see examples where $Z$ is an infinite metric space, a subspace of $\mathbb{R}$ of $\mathbb{R}^2$, or something like that. Thanks.