We explore the prospects of extending the LHC from a hadron-hadron to a photon-hadron collider and performing studies of inclusive quarkonium photoproduction. We perform an extrapolation from HERA data to predict the yields of photoproduced $J/\psi$ mesons in proton-lead collisions. Similarly, we perform an extrapolation from LHC data to model the large hadronic background. Inclusive photoproduction can be isolated from this background in the LHC detectors, namely, ALICE, ATLAS, CMS, and LHCb by imposing constraints on the hadronic activity in the main detector and in the far-forward region. This work focuses on the CMS detector.