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This paper proposes a three-degrees-of-freedom (3DOF) model to simulate the vibration of the vehicle and its chair, which is designed by using the quasi-zero-stiffness (QZS) vibration isolation mechanism. Firstly, the mathematical model of this system is established, for which the non-dimensionalized procedure is carried out. Next, the nonlinear dynamics of the 3DOF system is analyzed based upon the bifurcation theory, which demonstrates the bifurcation modes about the typical parameters of this system and reveals the parametric influences on the phenomena of the excitation resonances and internal resonances by exploiting the frequency-response analysis. This research will be helpful to improve the performance of the vehicle suspension and the comfort of the vehicle chair.