We report $^{125}$Te nuclear magnetic resonance and $^{181}$Ta nuclear quadrupole resonance studies on single-crystal Ta$_{4}$Pd$_{3}$Te$_{16}$, which has 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. By comparing the spin-lattice relaxation rate ( $1/T_{1}$) of $^{181}$Ta and $^{125}$Te, we found that electric-field-gradient (EFG) fluctuations develop below $80$ K. The EFG fluctuations are enhanced with decreasing temperature due to 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 the superconducting state, $1/T_{1}$ shows a Hebel-Slichter coherence peak just below $T_{rm c}$ for $^{125}$Te, indicating that Ta$_{4}$Pd$_{3}$Te$_{16}$ is a full-gap superconductor without nodes in the gap function. The coherence peak is absent in the $1/T_{1}$ of $^{181}$Ta due to the strong EFG fluctuations.
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