15x^2 will have 3 as a factor an odd number of times. This implies 15x^2 mod 9 = 3 or 6.

7y^2 will have 3 as a factor an even number of times or not at all. This implies that 7y^2 mod 9 can be any of 0, 1, 2, 4, 5, 7, or 8.

Trivially 9 mod 9 = 0.

There is no way to choose one number from {3,6} and one number from {0,1,2,4,5,7,8} whose difference is equal to 0 mod 9. Therefore the equation 15x^2-7y^2=9 has no integer solutions.