  We compute the relative-order-v^4 contribution to gluon fragmentation into
quarkonium in the 3S1 color-singlet channel, using the nonrelativistic QCD
(NRQCD) factorization approach. The QCD fragmentation process contains infrared
divergences that produce single and double poles in epsilon in 4-2epsilon
dimensions. We devise subtractions that isolate the pole contributions, which
ultimately are absorbed into long-distance NRQCD matrix elements in the NRQCD
matching procedure. The matching procedure involves two-loop renormalizations
of the NRQCD operators. The subtractions are integrated over the phase space
analytically in 4-2epsilon dimensions, and the remainder is integrated over the
phase-space numerically. We find that the order-v^4 contribution is enhanced
relative to the order-v^0 contribution. However, the order-v^4 contribution is
not important numerically at the current level of precision of
quarkonium-hadroproduction phenomenology. We also estimate the contribution to
hadroproduction from gluon fragmentation into quarkonium in the 3PJ color-octet
channel and find that it is significant in comparison to the complete
next-to-leading-order-in-alpha_s contribution in that channel.
