We theoretically study the spin multiplicity in the ground state of the square quantum dot (SQD) including four electrons, and discuss a possibility of the singlet-triplet (S/T) instability in the quantum system. As predicted by Hund's rule, the ground-state triplet appears when the DQD has a point group symmetry of D 4h. This ground-state triplet is also found even in the deformed SQD (D 2h) if the confinement length L is elongated. Consequently, the S/T instability is expected along these boundary lines. It is also worthwhile to notice that the present DFT as well as UHF calculation predicts the spin-singlet ground-state when the inter-electron interaction is strengthened (larger L) even though the SQD maintains its geometrical form of D 4h. The strong electron-localization causes the destabilization in the orbital (kinetic) energies, and produces this characteristic "anti-Hund" state.