How did you define product topology?
Anyway, you should know that a set $Q$ in the product topology of the product is open (in case of finitely many factors, but you have only two) if and only if to every $(x,y)\in Q$ there are open neighbourhoods $U_x, V_y$ of $x, y$ respectively such that $ U_x\times V_y \subset Q$. Since both factors are metric spaces this is true if and only if to each $(x,y) \in Q$ there are real numbers $r,s$ such that the product of the open balls is contained in $Q$, that is $B_r(x) \times B_s(y)\subset Q$. This implies that that the ball of radius $\min(r,s)$ is a subset of $Q$ with respect to the product metric.
There is only a little step missing, the other direction, when you start out with a set which open with respect to the product metric. You should finish this yourself.