  Inspired by the recent near-threshold $J/\psi$ photoproduction measurements,
we discuss gluon gravitational form factors (GFFs) and internal properties of
the proton. This work presents a complete analysis of the proton gluon GFFs
connecting the gluon part of the energy-momentum tensor and the heavy
quarkonium photoproduction. In particular, a global fitting of the $J/\psi$
differential and total cross section experimental data is used to determine the
gluon GFFs as functions of the squared momentum transfer $t$. Combined with the
quark contributions to the $D$-term form factor extracted from the deeply
virtual Compton scattering experiment, the total $D$-term is obtained to
investigate their applications in describing the proton mechanical properties.
These studies provide a unique perspective on investigating the proton gluon
GFFs and important information for enhancing QCD constraints on the gluon GFFs.
