Background: A nonperturbative charm production contribution, known as intrinsic charm, has long been speculated but has never been satisfactorily proven. The SeaQuest experiment at the Fermi National Accelerator Laboratory (FNAL) is in an ideal kinematic region to provide evidence of $J/\psi$ production by intrinsic charm. Purpose: $J/\psi$ production in the SeaQuest kinematics is calculated with a combination of perturbative quantum chromodynamics (QCD) and intrinsic charm to see whether the SeaQuest data can put limits on an intrinsic charm contribution. Methods: $J/\psi$ production in perturbative QCD is calculated to next-to-leading order in the cross section. Cold nuclear matter effects include nuclear modification of the parton densities, absorption by nucleons, and $p_T$ broadening by multiple scattering. The $J/\psi$ contribution from intrinsic charm is calculated assuming production from a $|uudc\overline{c}\rangle$ Fock state. Results: The nuclear modification factor $R_{pA}$ is calculated as a function of $x_F$ and $p_T$ for $p+C,p+\mathrm{Fe}$, and $p+W$ interactions relative to $p+d$. Previous data from the E866 experiment at FNAL is also shown and compared with the calculations. Conclusions: The SeaQuest kinematic acceptance is ideal for testing the limits on intrinsic charm in the proton.