  We report $^{75}$As NMR measurements on the new quasi one-dimensional
superconductor K$_{2}$Cr$_{3}$As$_{3}$ ($T_{c} \sim 6.1$~K) [J.\ K.\ Bao et
al., Phys. Rev. X {\bf 5}, 011013 (2015)]. We found evidence for strong
enhancement of Cr spin fluctuations above $T_c$ in the
[Cr$_{3}$As$_{3}$]$_{\infty}$ double-walled subnano-tubes based on the nuclear
spin-lattice relaxation rate $1/T_{1}$. The power law temperature dependence,
$1/T_{1}T \sim T^{-\gamma}$ ($\gamma \sim 0.25$), is consistent with the
Tomonaga-Luttinger liquid. Moreover, absence of the Hebel-Slichter coherence
peak of $1/T_{1}$ just below $T_{c}$ suggests unconventional nature of
superconductivity.
