I know that Fermi-Dirac statistics can be used to account for covalent bonding, but how does quantum mechanics explain hydrogen bonding?
In the case of covalent bondig, as for any other type of bonding, Quantum Mechanics, including the fermionic statistics of electrons, is part of the story. The other part is always the electrostatics resulting from the charge rearrangement.
The same applies to hydrogen bonding. Its general definition (IUPAC) implies that there should be a hydrogen atom, part of a more electronegative molecular fragment, thus exposing the positive charge of its nucleus to another molecule or molecular group, and polarizing it. The whole process of partial stripping of the electron on hydrogen and the polarization of the other group or molecule needs Quantum Mechanics for its explanation. The detailed accounting of the effect strongly depends on the specific molecules involved.