Galaxy quenching, the intricate process through which galaxies transition from active star-forming states to retired ones, remains a complex phenomenon that requires further investigation. This study investigates the role of active galactic nuclei (AGNs) in regulating star formation by analyzing a sample of 643 nearby galaxies with redshifts between 0.005 and 0.03 from the Calar Alto Legacy Integral Field Area (CALIFA) survey. Galaxies were classified according to the Quenching Stages and Nuclear Activity (QueStNA) scheme, which categorizes them based on their quenching stage and the presence of nuclear activity. We further utilized the integrated Extragalactic Database for Galaxy Evolution (iEDGE), which combined homogenized optical integral field unit and CO observations. This allowed us to examine how AGNs influence the molecular gas reservoirs of active galaxies compared to their non-active counterparts at similar evolutionary stages. Our Kolmogorov–Smirnov and χ2 tests indicate that the star formation property distributions and scaling relations of AGN hosts are largely consistent with those of non-active galaxies. However, AGN hosts exhibit systematically higher molecular gas masses across all quenching stages except for the quiescent nuclear ring stage. We find that AGN hosts follow the expected trends of non-active quenching galaxies, characterized by a lower star formation efficiency and molecular gas fraction compared to star-forming galaxies. Our results suggest that signatures of instantaneous AGN feedback are not prominent in the global molecular gas and star formation properties of galaxies.