  We investigate the relativistic Fermi motion effect in the case of $J/\psi$
production in various hadron colliders. A light-cone wave function is adopted
to represent the $J/\psi$ final state. The change in the confinement parameter
which sets a scale for the size of the final state, allows one to see the
effect in an explicit manner. While the effect has considerable influence on
the fragmentation probabilities and the differential cross sections, the total
cross sections are essentially left unchanged. such a feature is in agreement
with the momentum sum rule which the fragmentation functions should satisfy.
