We use photometric redshifts and statistical background subtraction to measure stellar mass functions in galaxy group-mass (4.5-8 x 10(13) M-circle dot) haloes at 1 < z < 1.5. Groups are selected from COSMOS and SXDF, based on X-ray imaging and sparse spectroscopy. Stellar mass (M-stell(ar)) functions are computed for quiescent and star-forming galaxies separately, based on their rest-frame UVJ colours. From these we compute the quiescent fraction and quiescent fraction excess (QFE) relative to the field as a function of M-stel(lar). QFE increases with M-st(ellar), similar to more massive clusters at 1 < z < 1.5. This contrasts with the apparent separability of M-stellar, and environmental factors on galaxy quiescent fractions at z similar to 0. We then compare our results with higher mass clusters at 1 < z < 1.5 and lower redshifts. We find a strong QFE dependence on halo mass at fixed M-ste(ll)ar; well fit by a logarithmic slope of d(QFE)/dlog (M-halo) similar to 0.24 +/- 0.04 for all M-stellar and redshift bins. This dependence is in remarkably good qualitative agreement with the hydrodynamic simulation BAHAMAS, but contradicts the observed dependence of QFE on M-stellar. We interpret the results using two toy models: one where a time delay until rapid (instantaneous) quenching begins upon accretion to the main progenitor ( 'no pre-processing') and one where it starts upon first becoming a satellite ('pre-processing'). Delay times appear to be halo mass-dependent, with a significantly stronger dependence required without pre-processing. We conclude that our results support models in which environmental quenching begins in low-mass (<10(14) M-circle dot) haloes at z > 1.