The excited state quantum dynamics of the organic cation in hybrid perovskites are investigated using the time-dependent density functional theory (TDDFT). The time-dependent non-adiabatic bond fluctuation behaviors reveal that the energy relaxation follows different pathways depending on the chemical bonding characteristics and energy transfer modes within the cation molecule, which can fundamentally affect its photostability. For the ammonium-group-containing cations, such as methylammonium (MA) or ethylammonium (EA), local vibrational modes survive for a long time. However, as their lowest unoccupied molecular orbital (LUMO) having π* characters, the amidinium-group-containing cations, such as formamidinium (FA) or guanidinium (GA), efficiently dissipate deposited energy via chaotic intramolecular vibrational energy redistribution (IVR). The distinct dynamic behaviors of A-site molecular cations are closely related to the quantum ergodicity, which can bring enhanced photochemical stability of FA and GA compared to MA and EA. Our theoretical investigation reveals the quantum chaos origin of better light stability of FA-based perovskites and serves the future research direction of the A-site engineering for better solar cells and light-emitting devices.