Simulating tunneling processes as well as their observation are challenging problems for many areas. In this study, we consider a double-well potential system coupled to a heat bath with a linear-linear (LL) and square-linear (SL) system-bath interactions. The LL interaction leads to longitudinal (T1) and transversal (T2) homogeneous relaxations, whereas the SL interaction leads to the inhomogeneous dephasing (T2*) relaxation in the white noise limit with a rotating wave approximation. We discuss the dynamics of the double-well system under infrared (IR) laser excitations from a Gaussian-Markovian quantum Fokker-Planck equation approach, which was developed by generalizing Kubo's stochastic Liouville equation. Analytical expression of the Green function is obtained for a case of two-state-jump modulation by performing the Fourier-Laplace transformation. We then calculate a two-dimensional infrared signal, which is defined by the four-body correlation function of optical dipole, for various noise correlation time, system-bath coupling parameters, and temperatures. It is shown that the bath-induced vibrational excitation and relaxation dynamics between the tunneling splitting levels can be detected as the isolated off-diagonal peaks in the third-order two-dimensional infrared (2D-IR) spectroscopy for a specific phase matching condition. Furthermore, this spectroscopy also allows us to directly evaluate the rate constants for tunneling reactions, which relates to the coherence between the splitting levels; it can be regarded as a novel technique for measuring chemical reaction rates. We depict the change of reaction rates as a function of system-bath coupling strength and a temperature through the 2D-IR signal.