Well, it's not elegant, but it works.

I'm using the definition that q is a quadratic residue (mod 35) if there exists an integer x, 0 < x < 35, such that x^2 = q (mod 35).

So I made a list of all such possible q. (Simply take all possible 0<x<18, the list repeats itself backward for 17<x<36, and find q for each x.) I get that q = 1, 4, 9, 11, 14, 15, 16, 21, 25, 29, 30 are all quadratic residues (mod 35).

A counter-example is 2 and 3. Both 2 and 3 are not quadratic residues (mod 35) and neither is 2 x 3 = 6.

-Dan