  We report resonant second harmonic generation (SHG) spectroscopy of an
hBN-encapsulated monolayer of MoS$_2$. By tuning the energy of the excitation
laser, we identify a dark state transition (D) that is blue detuned by +25 meV
from the neutral exciton X$^0$. We observe a splitting of the SHG spectrum into
two distinct peaks and a clear anticrossing between them as the SHG resonance
is tuned through the energy of the dark exciton D. This observation is
indicative of quantum interference arising from the strong two-photon
light-matter interaction. We further probe the incoherent relaxation from the
dark state to the bright excitons, including X$^0$ and localized excitons LX,
by the resonant enhancement of their intensities at the SHG-D resonance. The
relaxation of D to bright excitons is strongly suppressed on the bare substrate
whilst enabled when the hBN/MoS$_2$/hBN heterostructure is integrated in a
nanobeam cavity. The relaxation enabled by the cavity is explained by the
phonon scattering enhanced by the cavity phononic effects. Our work reveals the
two-photon quantum interference with long-lived dark states and enables the
control through nanostructuring of the substrate. These results indicate the
great potential of dark excitons in 2D-material based nonlinear quantum
devices.
