jmc

Let $p$ and $q$ be the roots. Then by Vieta's formulas, $p+q=63.$

If both $p$ and $q$ are odd, then $p+q$ is even, so one of $p$ or $q$ must be even. This means one of $p$ and $q$ is 2, and the other is $63-2=61.$ Therefore, $k=2\cdot 61=122,$ so there is only $\boxed{1}$ possible value of $k.$