Random illumination microscopy (RIM) could surpass the diffraction barrier in fluorescence microscopy by illuminating an object with unknown speckle patterns. It has been demonstrated that the resolution in RIM using second-order statistics is as good as that of conventional structured illumination microscopy (SIM) from the asymptotic point of view. Compared to classical SIM, RIM is more robust to optical aberrations and scattering introduced by thick samples. In this work, I show that the quantum correlations could further improve the resolution in random illumination microscopy due to the photon antibunching property of fluorophore emitters. In theory, the super-resolution capacity of this quantum-enhanced version of RIM corresponds to the fourth power of the point spread function under the general epi-illumination geometry.