We report a measurement of the νµ charged current quasi-elastic cross-sections on carbon in the T2K on-axis neutrino beam. The measured charged current quasi-elastic cross-sections on carbon at mean neutrino energies of 1.94 GeV and 0.93 GeV are (11.95 ± 0.19(stat.) +1.82−1.47(syst.)) ×10^−39 cm^2/neutron, and (10.64 ± 0.37(stat.)+2.03−1.65(syst.)) × 10^−39 cm^2/neutron, respectively. These results agree well with the predictions of neutrino interaction models. In addition, we investigated the effects of the nuclear model and the multi-nucleon interaction.