  We study the star formation histories (SFHs) and mass-weighted ages of 331
UVJ-selected quiescent galaxies in 11 galaxy clusters and in the field at
1<z<1.5 from the Gemini Observations of Galaxies in Rich Early ENvironments
(GOGREEN) survey. We determine the SFHs of individual galaxies by
simultaneously fitting rest-frame optical spectroscopy and broadband photometry
to stellar population models. We confirm that the SFHs are consistent with more
massive galaxies having on average earlier formation times. Comparing galaxies
found in massive clusters with those in the field, we find galaxies with
$M_\ast<10^{11.3}$ M$_{\odot}$ in the field have more extended SFHs. From the
SFHs we calculate the mass-weighted ages, and compare age distributions of
galaxies between the two environments, at fixed mass. We constrain the
difference in mass-weighted ages between field and cluster galaxies to
$0.31_{^{-0.33}}^{_{+0.51}}$ Gyr, in the sense that cluster galaxies are older.
We place this result in the context of two simple quenching models and show
that neither environmental quenching based on time since infall (without
pre-processing) nor a difference in formation times alone can reproduce both
the average age difference and relative quenched fractions. This is distinctly
different from local clusters, for which the majority of the quenched
population is consistent with having been environmentally quenched upon infall.
Our results suggest that quenched population in galaxy clusters at z>1 has been
driven by different physical processes than those at play at z=0.
