  We employ the projector quantum Monte Carlo simulations to study the
ground-state properties of the square-lattice SU(4) Hubbard model with a $\pi$
flux per plaquette. In the weak coupling regime, its ground state is in the
gapless Dirac semi-metal phase. With increasing repulsive interaction, we show
that, a Mott transition occurs from the semimetal to the valence bond solid,
accompanied by the $Z_4$ discrete symmetry breaking. Our simulations
demonstrate the existence of a second-order phase transition, which confirms
the Ginzburg-Landau analysis. The phase transition point and the critical
exponent $\eta$ are also estimated. To account for the effect of a $\pi$ flux
on the ordering in the strong coupling regime, we analytically derive by the
perturbation theory the ring-exchange term which describes the leading-order
difference between the $\pi$-flux and zero-flux SU(4) Hubbard models.
