When is the product of two quotient maps a quotient map?

Your weaker statement is almost true.

If $f: X \to Y$ is a quotient map and $Z$ is locally compact, then $f \times \operatorname{id}$ is a quotient map. I believe that this result is due to Whitehead.

More generally, if $f: X \to Y$ and $g: Z \to W$ are quotient maps and $Y$ and $Z$ are locally compact, then the product $f \times g: X \times Z \to Y \times W$ is a quotient map.

Why? Use the Whitehead theorem twice, since $f \times g = (\operatorname{id} \times g) \circ (f \times \operatorname{id})$.

See Munkres $\S 22$ for counterexamples.

@Ittay: In a cartesian closed category of spaces, a product of identification maps is an identification map. Here is a typical proof essentially from Topology and Groupoids p. 192.

It suffices to prove that if $f:Y \to Z$ is an identification map, then so also is $f \times 1: Y \times X \to Z \times X$ for any $X$.

Let $g:Z \times X \to W$ be a function such that $l=g(f \times 1):Y \times X \to W$ is continuous. By cartesian closedness, we have associated maps

$$ l': Y \to K(X,W), \quad g':Z \to K(X,W)$$

where $K(X,W)$ is the internal hom, and $g'f=l'$. Since $l'$ is continuous, and $f$ is an identification map, then $g'$ is continuous. Hence $g$ is continuous.

In the category of compactly generated spaces, I think that the product of quotient maps is (always) a quotient map.