  I compute the leading color-octet contributions to the process $\gamma + p
\to J/\psi + \gamma (+ X)$ within the non-relativistic QCD (NRQCD)
factorization formalism. In the color-singlet model, $J/\psi + \gamma$ can only
be produced when the photon interacts through its structure function, while the
color-octet mechanism allows for production of $J/\psi + \gamma$ via direct
photon-gluon fusion. Resolved photon processes can be easily be distinguished
from direct photon processes by examining the fraction of the incident photon
energy carried away by the $J/\psi$ in the event. Therefore, this process
provides a conclusive test of the color-octet mechanism. $J/\psi + \gamma$
production is particularly sensitive to the NRQCD matrix element which figures
prominently in the fragmentation production of $J/\psi$ at large $p_{\perp}$ in
hadron colliders. I also examine the predictions of the color evaporation model
(CEM) of quarkonium production and find that this process can easily
discriminate between the NRQCD factorization formalism and the CEM.
