Answer: 588

Small Hint:

Move a suitable multiple of $3x^2+1$ to create a product

Big Hint:

Factor $507$ and use that $3x^2+1\equiv1\pmod3$

Solution:

Rearranging gives $(3x^2+1)(y^2-10)=507,$ where $507=3\cdot13^2.$ The positive divisors of $507$ congruent to $1\pmod3$ are $1,$ $13,$ $169.$ If $3x^2+1=1,$ then $x^2=0$ but $y^2=517,$ which is not a square. If $3x^2+1=13,$ then $x^2=4$ and $y^2=49.$ Finally, $3x^2+1=169$ gives $x^2=56,$ not a square. Therefore $3x^2y^2=3\cdot4\cdot49=588.$