A nonperturbative charm production contribution, known as intrinsic
charm, has long been speculated but has never been satisfactorily proven. The
SeaQuest experiment at FNAL is in an ideal kinematic region to provide
evidence of $J/\psi$ production by intrinsic charm. Here, $J/\psi$ production
in the SeaQuest kinematics is calculated with a combination of perturbative
QCD and intrinsic charm to see whether the SeaQuest data can put limits on an
intrinsic charm contribution. $J/\psi$ production in perturbative QCD is
calculated to next-to-leading order in the cross section. Cold nuclear matter
effects included in this component are 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 $|uud c \overline c \rangle$ Fock state. The
nuclear modification factor, $R_{pA}$, is calculated as a function of $x_F$
and $p_T$ for $p+{\rm C}$, $p+{\rm Fe}$, and $p+{\rm W}$ interactions
relative to $p+{\rm d}$. It is shown that the SeaQuest kinematic acceptance
is ideal for setting limits on intrinsic charm in the proton.