  The study of exclusive processes in ultraperipheral collisions at the Large
Hadron Collider (LHC) has allowed us to test several aspects of the Standard
Model and to search for New Physics. In this letter, we investigate the
possibility of using these processes to improve our understanding of the
quarkonium production mechanism through the study of the exclusive $J/\Psi$
plus jet associate production in ultraperipheral $PbPb$ collisions. We estimate
the transverse - momentum and rapidity distributions considering that the
$\gamma \gamma \rightarrow J/\Psi + X$ ($X = \gamma, \, g$) subprocess is
described by the Non - Relativistic QCD (NRQCD) formalism and present
predictions for the rapidity ranges covered by central and forward detectors.
The experimental separation of these events is discussed and the results
indicate that a future experimental analysis is, in principle, feasible in
future runs of the LHC and the Future Circular Collider (FCC).
