Josephson plasmon in cuprate superconductors provides a direct fingerprint of c-axis superconducting coherence in high-T_c cuprate superconductors, manifesting itself as a plasma edge structure in the c-axis reflectivity spectrum typically in the terahertz frequency range. Unlike normal plasmon in metals, the Josephson plasmon possesses an intrinsic nonlinearity arising from the sine-Gordon equation that the interlayer phase differences obey. In this study, we performed two-dimensional terahertz(2D-THz) spectroscopy in La-based high-T_c cuprate superconductors, La_2-xSr_xCuO₄, to study the behavior of Josephson plasmon. In particular, we focus on the coherent nonlinear response that is distinct from the incoherent responses, e.g. arising from the incoherent quasiparticle excitations, and investigate the intrinsic nonlinearity associated with the Josephson plasmon. Based on the fundamental characterization of Josephson plasmon through the 2D-THz spectroscopy, we will discuss about its potential extension to the light-induced states.