Rb-NMR study has been performed on the quasi-one dimensional competing spin chain Rb 2 Cu 2 Mo 3 O 12 with ferromagnetic and antiferromagnetic exchange interactions on nearest neighboring and next nearest neighboring spins, respectively. The system changes from a gapped ground state at zero field to the gapless state at C ≃ 2 T, where the existence of magnetic order below 1 K was demonstrated by a broadening of NMR spectrum, associated with a critical divergence of 1 1 ⁄ . In higher temperature region, 1 −1 showed a power-law type temperature dependence, from which the field dependence of Luttinger parameter K was obtained and compared with theoretical calculations based on the spin nematic Tomonaga Luttinger Liquid (TLL) state.