H-1-NMR study of the organic radical compound BIPNNBNO which is a two-dimensional spin-gap system has been performed under the applied field up to 7.7 T including the critical field region and in the temperature down to 0.4 K. The proton spin-lattice relaxation rate T-1(-1) exhibited the thermal activation-type behavior suggesting the existence of the energy gap not only in the gapped state near zero field but also in the 1/3-plateau phase above about 6 T. It was found that T-1(-1) increased with decreasing temperature in the critical field region between the gapped phase and the plateau phase, which is similar to what has been already observed for one-dimensional systems associated with the Tomonaga-Luttinger liquid (TLL) state.