We report single-crystal V51 NMR studies on volborthite Cu3V2O7(OH)2·2H2O, which is regarded as a quasi-two-dimensional frustrated magnet with competing ferromagnetic and antiferromagnetic interactions. In the 1/3 magnetization plateau above 28 T, the nuclear spin-lattice relaxation rate 1/T1 indicates an excitation gap with a large effective g factor in the range of 4.6–5.9, pointing to magnon bound states. Below 26 T where the gap has closed, the NMR spectra indicate small internal fields with a Gaussian-like distribution, whereas 1/T1 shows a power-law-like temperature dependence in the paramagnetic state, which resembles a slowing down of spin fluctuations associated with magnetic order. We discuss the possibility of an exotic spin state caused by the condensation of magnon bound states below the magnetization plateau.