 Solution

The first excited unperturbed level is twofold degenerate, spanned by $|1,2⟩$ and $|2,1⟩$, with energy $5h^2/(8m{a}^2)$. The perturbation is $q{\epsilon }_Q(x-a/2)(y-a/2)$. Both diagonal matrix elements vanish, but the off-diagonal element does not:Diagonalizing the $2\times 2$ matrix with zero diagonal gives first-order shiftswith symmetric/antisymmetric combinations $(|1,2⟩\pm |2,1⟩)/\sqrt{2}$. Their order depends on the sign of $q{\epsilon }_Q$; the lower branch has shift $-|W_{12,21}|$. Applying nondegenerate perturbation theory to either product state would miss the splitting.