This is some old Olympiad problem (I vaguely remember its Iranian )
I factorised it as
$ (x-y)(3x + 3y +1) = y^2 $
Then I fruitlessly tried to prove
$\gcd(x-y, 3x + 3y +1 ) = \gcd (6x+1,6y+1) = 1$
for proving that $x-y$ is a perfect square.
How do I continue my method or are there other ways to approach this intuitively?