  We compute the contributions due to gluon and heavy-quark fragmentation to
quarkonium production at large transverse momentum in $\bar p p$ and $p p$
collisions. For inclusive $J/\psi$ production, there is a large contribution
from the $g \rightarrow \chi_c$ fragmentation. At large $p_T$, this is
comparable to the conventional charmonium model prediction via gluon fusion at
the ISR and dominates over the latter at the Tevatron energy. This may help to
explain, at least partly, the large $J/\psi$ production cross-section recently
observed by the CDF experiment. However, the fragmentation contribution to
$\psi^{\prime}$ production is not large enough to explain the corresponding CDF
data. We also present the results for $\Upsilon$ production at ISR and Tevatron
energies.

High Energy Physics - Phenomenology