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 broad-band 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-* < 10(11.3) M-circle dot 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.