  We develop a density-dependent quark mean-field (DDQMF) model to study the
properties of nuclear matter and neutron stars, where the coupling strength
between $\sigma$ meson and nucleon is generated by the degree of freedom of
quarks, while other meson coupling constants are regarded as density-dependent
ones. Two values for the nucleon effective mass, $M^*_{N0}/M_N=0.556,~0.70$ at
the saturation density are chosen based on the consideration of the
core-collapse supernova simulation and finite nuclei when the meson-nucleon
coupling constants are fixed. We find that the equation of state (EOS) of
nuclear matter, the symmetry energy, the mass-radius relations, and the tidal
deformabilities of neutron stars with larger nucleon effective mass are more
sensitive to the skewness coefficient $J_0$. The EOSs with $M^*_{N0}/M_N=0.70$
are softer when the skewness coefficient $J_0=-800$ MeV. However, the maximum
masses of the neutron star can be around $2.32M_\odot$ with $J_0=400$ MeV
regardless of the value of the nucleon effective mass. By manipulating the
coupling strength of the isovector meson to generate different slopes of
symmetry energy, we construct the neutron star EOSs that can satisfy the
different variables from the simultaneous mass-radius measurements of PSR
J0030+0451, PSR J0740+6620 by the NICER collaboration, the mass-radius
relations of HESS J1731-347, and the radius constraints from the
gravitational-wave signal GW170817 in the framework of DDQMF model. At the same
time, most of these constructed EOSs can also satisfy the constraints of the
tidal deformability from GW170817 event.
