<p>The kagomé lattice, characterized by its unique geometry, hosts exotic electronic structures such as Dirac points, flat bands, and van Hove singularities, which are closely linked to electronic instabilities. Recently, the <i>A</i>V₃Sb₅ family (<i>A</i> = K, Rb, Cs) has emerged as a prime platform for exploring these intriguing phenomena, exhibiting both charge-density wave (CDW) and superconductivity. The coexistence of two phenomena suggests the possibility of unconventional superconducting mechanisms. Here, we report a detailed investigation of two phenomena in CsV₃Sb₅ using high-quality single crystals. For zero-field ¹²¹Sb-nuclear quadrupole resonance, a Hebel-Slichter peak suggesting the realization of conventional superconductivity is observed just below the superconducting transition temperature. In contrast, ⁵¹V-nuclear magnetic resonance shows an anomalous increase in spin fluctuations deep inside the superconducting phase, suggesting the unconventional properties in CsV₃Sb₅ superconductivity. Our work enhances the understanding of unconventional superconducting properties in kagomé lattice systems by revealing the relationship between two phenomena.</p>