We measured the 13C-NMR spectrum and T1 of the quasi-two-dimensional organic superconductor kappa-(BEDT-TTF)2Cu(NCS)_{2} under pressure. This material was thought to show a relationship between T_{c} and the effective cyclotron mass m_{c};{*}, obtained from the Shubnikov-de Haas (SdH) effect. We found that kappa-(BEDT-TTF)2Cu(NCS)_{2} behaved as a Fermi liquid at low temperature under all pressures, and antiferromagnetic fluctuations were expected. The pressure dependence of the Korringa factor is similar to that of the effective cyclotron mass m_{c};{*}, suggesting that antiferromagnetic fluctuations contribute to the superconductivity of this material. We also found that, under pressure, T;{*} was shifted to 150 K, the temperature characteristic of the shift from bad metal to good metal.