If we take a look at the expression for the perturbing Hamiltonian, we clearly see that the contribution from EFG tensor is present through Vzz and eta. Knowing that eta ~ Vxx-Vyy and that our toy model (its quantum allowed states at least) is equivalent along x and y axis, we should end up with eta = 0, so only the Vzz component is present.

The rotational symmetry around the z axis in the quantum case is the source of the axial symmetry in the principle axis system.