We have investigated the carrier spin relaxation in GaInNAsSb/GaNAsSb/GaAs quantum well (QW) by time-resolved photoluminescence (PL) measurement. The sample consists of an 8-nm-thick GaIn 0.36N 0.006AsSb 0.015 well, 5-nm-thick GaN 0.01AsSb 0.11 intermediate barriers and 100-nm-thick GaAs barriers grown by molecular beam epitaxy on a GaAs(100) substrate. The spin relaxation time and recombination lifetime at 10 K are measured to be 228 ps and 151 ps, respectively. As a reference, we have also obtained a spin relaxation time of 125 ps and a recombination lifetime of 63 ps for GaInNAs/GaNAs/GaAs QW. This result shows that crystal quality is slightly improved by adding Sb, although these short carrier lifetimes mainly originate from a nonradiative recombination. These spin relaxation times are longer than the 36 ps spin relaxation time of InGaAs/InP QWs and shorter than the 2 ns spin relaxation time of GaInNAs/GaAs QW.