A method that includes explicit electron-proton correlation directly into the nuclear-electronic orbital self-consistent-field framework is presented. This nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) scheme is formulated using Gaussian basis functions for the electrons and the quantum nuclei in conjunction with Gaussian-type geminal functions. The NEO approach is designed for the quantum treatment of a relatively small number of nuclei, such as the hydrogen nuclei involved in key hydrogen bonding interactions or hydrogen transfer reactions. The conventional nuclear-electronic-orbital-based methods produce nuclear wave functions that are too localized, leading to severe overestimations of hydrogen vibrational frequencies, as well as inaccuracies in geometries, isotope effects, couplings, and tunneling splittings. The application of the NEO-XCHF approach to a model system illustrates that the description of the nuclear wave function is significantly improved by the inclusion of explicit electron- proton correlation. In contrast to the NEO-HF method, the NEO-XCHF method leads to hydrogen vibrational stretch frequencies that are in excellent agreement with those calculated from grid-based methods. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically and only electron-proton correlation is treated explicitly. Electron-electron dynamical correlation can be included with density functional theory or perturbation theory methods.