[en] RbGd₂Fe₄As₄O₂ is a newly discovered self-hole-doped stoichiometric superconductor, which has a hybrid structure with separated double FeAs layers and exhibits a high superconducting transition temperature T _c  =  35 K. Here, we report the effect of pressure (P) on its T _c and normal-state transport properties by measuring the temperature dependence of resistivity ρ(T) under various pressures up to 14 GPa with a cubic anvil cell apparatus. We found that the T _c is suppressed monotonically to ca. 12.5 K upon increasing pressure to 14 GPa with a slope change of T _c(P) at around 4 GPa. In addition, the low-temperature normal-state ρ(T), which is proportional to T ⁿ, also evolves gradually from a non-Fermi-liquid with n  =  1 at ambient pressure to a Fermi liquid with n  =  2 at P  ⩾  4 GPa. Accompanying with the non-Fermi-liquid to Fermi-liquid crossover, the quadratic temperature coefficient of resistivity, which reflects the effective mass of charge carriers, also experiences a significant reduction as commonly observed in the vicinity of a magnetic quantum critical point (QCP). Our results indicate that the stoichiometric RbGd₂Fe₄As₄O₂ at ambient pressure might be located near a QCP such that the enhanced critical spin fluctuations lead to high-T _c superconductivity. The application of pressure should broaden the electronic bandwidth and weaken the spin fluctuations, and then restore a Fermi-liquid ground state with lower T _c. (paper)