The dissipative vibrational dynamics of a symmetric triatomic ABA molecule is extensively studied in a series of three papers. The momentum-dependent rotor model is used to describe the dynamical behavior of an ABA molecule with a single 1:1 nonlinear resonant coupling. Four characteristic modes, including symmetric normal modes, asymmetric normal modes, local modes, and quasi-local modes, are identified at different energy regimes. Under the influence of thermal noise, an ABA molecule switches between the four distinct modes intermittently along its dynamic trajectory, resulting in blinking of vibrational bond energy. The statistics of these dynamic transitions (i.e., "vibrational blinking") is obtained from numerical integration of stochastic equations in this paper and will be analyzed within the theoretical framework of noise-induced symmetry breaking and activated barrier crossing in the next two papers. This study demonstrates the rich dynamic behavior of a highly nonlinear system in a dissipative environment and is instructive for understanding the stability and stochasticity of energy localization in complex systems ranging from polyatomic molecules to nanostructures and micromechanical oscillators.