The electric field gradient tensor is given in the question sheet. In the slides, we have seen that this tensor is diagonal in its principle axis system (I think this is actually the definition of PAS: a system for which this tensor is diagonal. In general, given the tensor, we should diagonalise it, which gives us a matrix decomposition dictating a change of basis, rotating us into a frame in which the tensor IS diagonal, and this is the PAS). So by chance, or by luck, we chose the correct system in which this is already diagonal.
Looking at the visualisation, it is clear that there is axial symmetry. The symmetry axis is the z-axis. Indeed, we see that the toy model is symmetric with respect to this axis. For instance, the negative charges both lie on this axis, and are at the same distance away from the origin. Also the ‘nucleus’, the dumbbell, is symmetric with respec to this axis. Hence the full toy model has axial symmetry wrt the z-axis.