We examine critical properties of the quarter-filled one-dimensional Hubbard model with dimerization and with the onsite and nearest-neighbor Coulomb repulsion U and V. By utilizing the bosonization method, it is shown that the system exhibits an Ising quantum phase transition from the Mott insulating state to the charge-ordered insulating state. It is also shown that the dielectric permittivity exhibits a strong enhancement as decreasing temperature with power-law dependence at the Ising critical point.