  We study the possibility of the existence of a deconfined quark matter in the
core of neutron star (NS)s and its relation to non-radial oscillation modes in
NSs and hybrid star (HS)s. We use relativistic mean field (RMF) models to
describe the nuclear matter at low densities and zero temperature. The
Nambu--Jona-Lasinio (NJL) model is used to describe the quark matter at high
densities and zero temperature. A Gibbs construct is used to describe the
hadron-quark phase transition (HQPT) at large densities. Within the model, as
the density increases, a mixed phase (MP) appears at density about $2.5$ times
the nuclear matter saturation density $(\rho_0)$ and ends at density about $5
\rho_0$ beyond which the pure quark matter phase appears. It turns out that a
stable HS of maximum mass, $M=2.27 M_{\odot}$ with radius $R=14$ km (for NL3
parameterisation of nuclear RMF model), can exist with the quark matter in the
core in a MP only. HQPT in the core of maximum mass HS occurs at radial
distance, $r_c=0.27R$ where the equilibrium speed of sound shows a
discontinuity. Existence of quark matter in the core enhances the non-radial
oscillation frequencies in HSs compared to NSs of the same mass. This
enhancement is significantly large for the $g$ modes. Such an enhancement of
the $g$ modes is also seen for a density dependent Bayesian (DDB)
parmeterisation of the nucleonic EOS. The non-radial oscillation frequencies
depend on the vector coupling in the NJL model. The values of $g$ and $f$ mode
frequencies decrease with increase the vector coupling in quark matter.
