Natural gas hydrate (NGH) is a kind of unique resource among hydrocarbon fuels. It is stable within its sedimentary hosts and will only convert to its constituent gas and water phases relatively quickly if reservoir pressure-temperature conditions are artificially disturbed. The unwanted vibrations from the drill string with the complex external environment and excitation usually cause instability in the NGH. In this paper, a nonlinear mathematical model is presented taking into account the fluid−solid coupling vibration caused by the drilling fluid inside and outside the drill string, the non-smooth rub-impact between the drill string and borehole wall, and the influence of centralizer setting interval on nonlinear resonant vibration of the drill string. The relationship between the variable parameters of the drill string and the complex dynamics of the drill string is analyzed. Finally, the bifurcation diagram is given. The results show that the system response exhibits complex dynamic behavior due to rub-impact and nonlinear effects. However, there is always a suitable parameter in the bifurcation diagram of each parameter change, so that the amplitude of the system is small and the failure probability of the drill string is reduced. The research results have reference significance for exploring the nonlinear dynamic mechanism of a drill string under complex working conditions and reasonably optimizing drilling parameters.