Your system is of degree 8, with 8 parameters. If the parameters are replaced with numbers, as in
With[{d = 3, A = 1, B = 2, F = 1,
G = 1, H = 1, J = 1, c = 1},
Solve[x^2 + y^2 + z^2 == 1 &&
2*x*y*1/Sqrt [d - 1] +
y^2*(d - 2)/(d - 1) + z^2 == 0 &&
x^2*A^2 + y^2*B^2 + z^2*F^2 + x*y*2*G +
x*z*2*H + y*z*2*J == c,
{x, y, z}, Reals]]
it is solved very quickly. Unfortunately, the parameters make everything complicated. Look at what happens by leaving the parameter c
free:
With[{d = 3, A = 1, B = 2, F = 1,
G = 1, H = 1, J = 1},
Solve[x^2 + y^2 + z^2 == 1 &&
2*x*y*1/Sqrt [d - 1] +
y^2*(d - 2)/(d - 1) + z^2 == 0 &&
x^2*A^2 + y^2*B^2 + z^2*F^2 + x*y*2*G +
x*z*2*H + y*z*2*J == c,
{x, y, z}, Reals]]
the solution is an expression that takes 2.5 MB to store, and it hard to understand and use. With 8 free parameters the situation is hopeless.