Diamond has attracted attention as a next-generation semiconductor because of its various exceptional properties such as a wide bandgap and high breakdown electric field. Diamond field-effect transistors, for example, have been extensively investigated for high-power and high-frequency electronic applications. The quality of their charge transport (i.e., mobility), however, has been limited due to charged impurities near the diamond surface. Here, we fabricate diamond field-effect transistors by using a monocrystalline hexagonal boron nitride as a gate dielectric. The resulting high mobility of charge carriers allows us to observe quantum oscillations in both the longitudinal and Hall resistivities. The oscillations provide important information on the fundamental properties of the charge carriers, such as effective mass, lifetime, and dimensionality. Our results indicate the presence of a high-quality two-dimensional hole gas at the diamond surface and thus pave the way for studies of quantum transport in diamond and the development of low-loss and high-speed devices.