\(^{1}\)H NMR and static susceptibility measurements have been performed in an organic Mott insulator with nearly isotropic triangular lattice, \(\kappa\)-(BEDT-TTF)\(_{2}\)Cu\(_{2}\)(CN)\(_{3}\), which is a model system of frustrated quantum spins. The static susceptibility is described by the spin \(S\) = 1/2 antiferromagnetic triangular-lattice Heisenberg model with the exchange constant \(J\) \(\sim\) 250 K. Regardless of the large magnetic interactions, the \(^{1}\)H NMR spectra show no indication of long-range magnetic ordering down to 32 mK, which is four-orders of magnitude smaller than \(J\). These results suggest that a quantum spin liquid state is realized in the close proximity of the superconducting state appearing under pressure.