We perform path integral molecular dynamics (PIMD) simulations of a monatomic liquid that exhibits a liquid-liquid phase transition (LLPT) and liquid-liquid critical point. PIMD simulations are performed using different values of the Planck's constant h, allowing us to study the behavior of the liquid as nuclear quantum effects (NQE, i.e., atoms delocalization) are introduced, from the classical liquid ( h=0) to increasingly quantum liquids ( h>0). By combining the PIMD simulations with the ring-polymer molecular dynamics (RPMD) method, we also explore the dynamics of the classical and quantum liquids. We find that (i) the glass transition temperature of the low-density liquid (LDL) is anomalous, i.e., TgLDL(P) decreases upon compression. Instead, (ii) the glass transition temperature of the high-density liquid (HDL) is normal, i.e., TgHDL(P) increases upon compression. (iii) NQE shift both TgLDL(P) and TgHDL(P) towards lower temperatures but NQE are more pronounced on HDL. We also study the glass behavior of the ring-polymer systems associated to the quantum liquids studied (via the PI formulation of statistical mechanics). There are two glass states in all the systems studied, LDA and HDA, which are the glass counterpart of LDL and HDL. In all cases, the pressure-induced LDA-HDA transformation is sharp, reminiscent of a first-order phase transition. In the low-quantum regime, the LDA-HDA transformation is reversible, with identical LDA forms before the compression and after the decompression. However, in the high-quantum regime, the atoms become more delocalized in the final LDA than in the initial LDA, raising questions on the reversibility of the LDA-HDA transformation.