A generation, propagation, and transfer of phonon angular momenta are examined on thermal transport in chiral insulative and diamagnetic crystals of α-quartz. We found that thermally driven phonons carry chirality-dependent angular momenta in the quartz crystals and they could be extracted from the quartz as a spin signal. Namely, chirality-induced selectivity of phonon angular momenta is realized in the chiral quartz. We argue that chiral phonons available in chiral materials could be a key element in triggering or enhancing chirality-induced spin selectivity with robust spin polarization and long-range spin transport found in various chiral materials.