We extend the quark mean-field (QMF) model for nuclear matter and study the possible presence of quark matter inside the cores of neutron stars. A sharp first-order hadron-quark phase transition is implemented combining the QMF for the hadronic phase with “constant-speed-of-sound” parametrization for the high-density quark phase. The interplay of the nuclear symmetry energy slope parameter, L, and the dimensionless phase transition parameters (the transition density , the transition strength , and the sound speed squared in quark matter ) are then systematically explored for the hybrid star proprieties, especially the maximum mass M_max and the radius and the tidal deformability of a typical 1.4M_⊙ star. We show the strong correlation between the symmetry energy slope L and the typical stellar radius R1.4, similar to that previously found for neutron stars without a phase transition. With the inclusion of phase transition, we obtain robust limits on the maximum mass () and the radius of 1.4M_⊙ stars (), and we find that a too-weak () phase transition taking place at low densities is strongly disfavored. We also demonstrate that future measurements of the radius and tidal deformability of ∼1.4M_⊙ stars, as well as the mass measurement of very massive pulsars, can help reveal the presence and amount of quark matter in compact objects.