It is a true or false question from an old test.
At first I tried some counterexamples, using the line with two origins or taking $B$ as a quotient space of the real line by some not-open subset, since I know the result is true if $B$ and $C$ are also Hausdorff spaces. It did not work.
I then tried to prove it instead, showing that $(p\times q)^{-1}(U)$ open implies $U$ open, but I can't work out how to use both hypothesis, or any other way to reach the answer.
Thanks for the help!
It also works for locally compact spaces which are not Hausdorff. If $Z$ is a locally compact space and $p:X\to Y$ is a quotient map, then $p\times\mathbf 1_Z:X\times Z\to Y\times Z$ is a quotient map. One way to show this is by verifying the universal property of quotient maps, and here you have to use the adjunction $\mathbf{Top}(Y\times Z,W)\cong\mathbf{Top}\left(Y,W^Z\right)$, which holds when $Z$ is locally compact. One can also show directly that $p\times\mathbf 1_Z$ is a quotient map, this is done by Ronald Brown in his book Topology and Groupoids on page 109.