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We report In-NQR and Co-NMR experiments of CeCoIn$_5$ that undergoes a superconducting transition with a record high $T_{\rm c}$ = 2.3 K to date among heavy-fermion superconductors. At zero magnetic field, an anomalous temperature ($T$) dependence of nuclear spin-lattice relaxation rate $1/T_1$ of $^{115}$In is explained by the relation $1/T_1\propto T\cdot\chi_Q(T)^{3/4}$ based on the anisotropic spin-fluctuations model in case of the proximity to an antiferromagnetic (AFM) quantum critical point (QCP). The novel behavior of $1/T_1\sim T^{1/4}$ over a wide $T$ range of $T_{\rm c} < T < 40$ K arises because the staggered susceptibility almost follows the Curie law $\chi_Q(T)\propto 1/(T+\theta)$ with $\theta= 0.6$ K and hence $1/T_1\propto T/(T+0.6)^{3/4}\sim T^{1/4}$ for $\theta < T$. We highlight that the behavior $1/T_1\sim T^{1/4}$ is due to the proximity to the anisotropic AFM QCP relevant with its layered structure, and is not associated with the AFM QCP for isotropic 3D systems. We have also found that the AFM spin fluctuations in CeCoIn$_5$ are suppressed by small magnetic field so that $\theta=0.6$ K at $H$=0 increases to $\theta = 2.5$ K at $H$ = 1.1 T, reinforcing that CeCoIn$_5$ is closely located at the QCP.