  We perform the first study of the principal core $g$-mode oscillation in
hybrid stars containing quark matter, utilizing a crossover model for the
hadron-to-quark transition inspired by lattice QCD. The ensuing results are
compared with our recent findings of $g$-mode frequencies in hybrid stars with
a first-order phase transition using Gibbs constructions. We find that models
using Gibbs construction yield $g$-mode amplitudes and the associated
gravitational energy radiated that dominate over those of the chosen crossover
model owing to the distinct behaviors of the equilibrium and adiabatic sound
speeds in the various models. Based on our results, we conclude that were
$g$-modes to be detected in upgraded LIGO and Virgo detectors it would indicate
a first-order phase transition akin to a Gibbs construction.
