We report a novel insulator-insulator transition arising from the internal charge degrees of freedom in the two-dimensional quarter-filled organic salt β-(meso-DMBEDT-TTF)2PF6. The optical conductivity spectra above Tc=70 K display a prominent feature of the dimer Mott insulator, characterized by a substantial growth of a dimer peak near 0.6 eV with decreasing temperature. The dimer peak growth is rapidly quenched as soon as a peak of the charge order appears below Tc, indicating a competition between the two insulating phases. Our infrared imaging spectroscopy has further revealed a spatially competitive electronic phase far below Tc, suggesting a nature of quantum phase transition driven by material-parameter variations.