xdog has proved that x and y cannot both be multiples of 3, but there are two additional steps needed to prove that they cannot be any other integral values:

Case 2) If y is divisible by 3 and x is not, then

15x^2 = 7y^2 + 9

the right hand side is divisible by 9, but the left hand is not, so there are no integral solutions.

Case 3) If y is not divisible by 3 then, y^2 (mod 3) = 1.

15x^2 = 7y^2 + 9, when reduced to mod 3,

becomes 0 = 1, so there are no solutions