  By taking into account the latest observations and theoretical constraints,
we investigate the merger and post-merger of binary neutron stars (NSs) with
general-relativistic numerical simulations employing hadronic and hybrid
equations of state (EOSs). We name our hybrid stars neutron-quark stars (NQS),
because the transition from hadrons to quarks starts at a density lower than
the central density of $\sim 1 M_{\odot}$ stars. We address two viable
scenarios for the transition to quark matter: a crossover or a strong
first-order phase transition (1PT). We find that a crossover transition is in
principle observable when both the inspiral and post-merger signals are
detected because the post-merger gravitational-wave (GW) main frequency $f_2$
is generally lower than that of hadronic models with the same tidal
deformability ($\Lambda$). Since it is viable according to current
multi-messenger constraints, we also highlight the possibility of an EOS with a
strong 1PT that takes place at 1.8 times the nuclear saturation density, with a
stiff quark EOS after the transition. It is the first time that mergers of
binary NQSs with a deconfined quark-matter core are studied numerically in full
general relativity. In this case, although ($\Lambda$, compactness) lies
significantly outside the hadronic relation, ($\Lambda$, $f_2$) is close to the
relation valid for hadronic EOSs. We also point out a linear correlation, valid
within the observational constraints and not sensitive to the presence of a
hadron-quark transition, between the emitted energy in GWs and their frequency.
