  We study quantum criticality of spinless fermions on the quasi one
dimensional $\pi$-flux square lattice in cylinder geometry, by using the
infinite density matrix renormalization group and abelian bosonization. For a
series of the cylinder circumferences $L_y=4n+2=2, 6, \cdots$ with the periodic
boundary condition, there are quantum phase transitions from gapped Dirac
fermion states to charge density wave (CDW) states. We find that the quantum
phase transitions for such circumferences are continuous and belong to the
(1+1)-dimensional Ising universality class. On the other hand, when $L_y=4n=4,
8, \cdots$, there are gapless Dirac fermions at the non-interacting point and
the phase transition to the CDW state is Gaussian. Both of these two
criticalities are described in a unified way by the bosonization. We clarify
their intimate relationship and demonstrate that a central charge $c=1/2$ Ising
transition line arises as a critical state of an emergent Majorana fermion from
the $c=2$ Gaussian transition point.
