Clarifying the interplay between charge-density waves (CDWs) and superconductivity is important in the kagome metal CsV3Sb5, and pressure (P) can play a crucial role. Here, we present 121/123Sb nuclear quadrupole resonance (NQR) measurements under hydrostatic pressures up to 2.43 GPa in CsV3Sb5 single crystals. We demonstrate that the CDW gradually changes from a commensurate modulation with a star-of-David (SoD) pattern to an incommensurate one with a superimposed SoD and Tri-hexagonal (TrH) pattern stacking along the c-axis. Moreover, the linewidth δν of 121/123Sb-NQR spectra increases with cooling down to TCDW, indicating the appearance of a short-range CDW order due to CDW fluctuations pinned by quenched disorders. The δν shows a Curie–Weiss temperature dependence and tends to diverge at Pc ~ 1.9 GPa, suggesting that a CDW quantum critical point (QCP) exists at Pc where Tc shows the maximum. For P > Pc, spin fluctuations are enhanced when the CDW is suppressed. Our results suggest that the maximal Tc at Pc ~ 1.9 GPa is related to the CDW QCP, and the presence of spin fluctuations prevents the Tc from a rapid decrease otherwise, after the CDW is completely suppressed.