From the expression for H_qq^nuc, one is supposed to prove the axial symmetry of the electric-field gradient tensor in its principle axis system. H_qq^nuc only features Vzz in its expression, no Vxx or Vyy. So my reasoning would be that the axial symmetry is due to the perturbance in only one direction, related to Vzz.
From the visualisation of the toy model, the symmetry is obvious. If you look at the picture, imagine the axis and the symmetry follows.