  The detection of gravitational waves from GW170817 has provided a new
opportunity to constrain the equation of state (EOS) of neutron stars. In this
article, we investigate the possible existence of quarks inside the neutron
star core in the context of GW170817. The nucleon phase is treated within the
relativistic nuclear mean-field approach where we have employed a fully
comprehensive set of available models, and the quark phase is described in the
Bag model. We show that the nucleonic EOSs which are inconsistent with the
tidal deformability bound become consistent when phase transition to quark
matter via Gibbs construction is allowed. We find that several nucleonic EOSs
support the presence of pure quark matter core with a small mass not more than
$0.17M_\odot$ confined within a radius of 0.9 km. We also find that the strong
correlation between tidal deformability and neutron star radii observed for
pure nucleonic stars does persist even with a nucleon-quark phase transition
and provides an upper limit on the radius of $R_{1.4} \lesssim 12.9$ km for a
$1.4M_\odot$ neutron star.
