  We study the principal core g-mode oscillation in hybrid stars containing
quark matter and find that they have an unusually large frequency range
($\approx$ 200 Hz - 1 kHz) compared to ordinary neutron stars or self-bound
quark stars. Theoretical arguments and numerical calculations that trace this
effect to the difference in the behaviour of the equilibrium and adiabatic
sound speeds in the mixed phase of quarks and nucleons are provided. We propose
that the unique sensitivity of core g-mode oscillations to the presence of a
mixed quark-nucleon phase can be an indicator of quark matter in neutron stars.
Based on our analysis, we conclude that for binary mergers where one or both
components may be a hybrid star, the fraction of tidal energy pumped into
resonant g-modes in hybrid stars can exceed that of a normal neutron star by a
factor of 2-3, although resonance occurs during the last stages of inspiral. A
self-bound star, on the other hand, has a much weaker tidal overlap with the
g-mode The cumulative tidal phase error in hybrid stars, $\Delta\phi\cong$ 0.5
radians, is comparable to that from tides in ordinary neutron stars, presenting
a challenge in distinguishing between the two cases. However, should the
principal g-mode be excited to sufficient amplitude for detection in a
post-merger remnant with quark matter in its interior, its distinct frequency
would be a strong and direct indication for the existence of hybrid stars.
