jmc

If we move the $x^2$ term to the left side, it is factorable: $$(3x^2+1)(y^2-10)=517-10$$ $507$ is equal to $3\cdot 13^2$. Since $x$ and $y$ are integers, $3x^2+1$ cannot equal a multiple of three. $169$ doesn't work either, so $3x^2+1=13$, and $x^2=4$. This leaves $y^2-10=39$, so $y^2=49$. Thus, $3x^2{y}^2=3\times 4\times 49=\boxed{588}$.