The cavity magnomechanical (CMM) system offers a new platform for studying macroscopic quantum effects. In previous studies of CMM systems, quantum effects have mainly been studied in the resolved-sideband regime, which presents challenges for experimental implementation. Here, we propose a scheme to enhance quantum effects in the CMM system by applying a Floquet field to periodically modulate the magnon mode. To verify the effectiveness of the scheme, we examine several key phenomena, including quantum entanglement, mechanical squeezing, and ground-state cooling of the mechanical mode. More importantly, with the assistance of periodic modulation, these phenomena can be observed in the unresolved-sideband regime. This relaxes the requirements for the implementation of quantum effects in experimental CMM systems. Our proposed periodic modulation mechanism for CMM systems is quite general and can also be applied to enhance and generate other quantum effects in the unresolved-sideband regime.