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We report $^{125}$Te nuclear magnetic resonance and $^{181}$Ta nuclearquadrupole resonance studies on single-crystal Ta$_{4}$Pd$_{3}$Te$_{16}$, whichhas a quasi-one-dimensional structure and superconducts below $T_{\rm c}=4.3$K. $^{181}$Ta with spin $I=7/2$ is sensitive to quadrupole interactions, while$^{125}$Te with spin $I=1/2$ can only relax by magnetic interactions. Bycomparing the spin-lattice relaxation rate ( $1/T_{1}$) of $^{181}$Ta and$^{125}$Te, we found that electric-field-gradient (EFG) fluctuations developbelow $80$ K. The EFG fluctuations are enhanced with decreasing temperature dueto the fluctuations of a charge density wave that sets in at $T_{\rm CDW}=20$K, below which the spectra are broadened and $1/T_{1}T$ drops sharply. In thesuperconducting state, $1/T_{1}$ shows a Hebel-Slichter coherence peak justbelow $T_{\rm c}$ for $^{125}$Te, indicating that Ta$_{4}$Pd$_{3}$Te$_{16}$ isa full-gap superconductor without nodes in the gap function. The coherence peakis absent in the $1/T_{1}$ of $^{181}$Ta due to the strong EFG fluctuations.

Charge fluctuations and nodeless superconductivity in quasi-one-dimensional Ta$_{4}$Pd$_{3}$Te$_{16}$ revealed by $^{125}$Te-NMR and $^{181}$Ta-NQR