Full quantum-scattering calculations are reported for low-energy near-threshold inelastic collision cross sections for $\mathrm{H}+\mathrm{Na}$. The calculations include transitions between all levels up to and including the ionic state (ion-pair production) for collision energies from the threshold up to 10 eV. These results are important for astrophysical modeling of spectra in stellar atmospheres. Results for the $3s\ensuremath{-}3p$ excitation are carefully examined using three different quantum chemistry input data sets, and large differences are found near the threshold. The differences are found to be predominantly due to differences in the radial coupling rather than potentials and are also found not to relate to differences in couplings in a simple manner. In fact, of the three input couplings, the two that are most similar give the cross sections with the largest differences. The $3s\ensuremath{-}3p$ cross sections show orbiting resonances which have been seen in earlier studies, while Feshbach resonances associated with closed channels were also found to be present in the low-energy cross sections for some transitions.