There exists a kind of quadratic nonlinear system with specific type of turbulent fluctuation excitation in nature, which belongs to a special non-Gaussian input signal system. Its characteristic is that the input signal spectrum is generated by turbulent fluctuations, and the power spectrum distribution of this turbulence fluctuation signal is close to Gaussian distribution. Starting with the work of Choi et al. ( 1985 J. Sound Vib . 99309 ) and Kim et al. [ 1987 IEEE J. Ocean. Eng. OE-12568 ), we extend the simulation of a specific turbulent fluctuation excited response-type quadratic nonlinear system represented by the wave excited mooring ship response, and fully develop the internal development of turbulence based on bispectral analysis technology. We also extend the simulation system and conduct systematic modeling analysis. The complete iterative method [ 2020 Phys. Scr. 95055202 ] is used to solve the model, and calculate the linear transfer function and quadratic nonlinear transfer function. The comparison of simulation and modeling results with the real systems and their models confirms the correctness of the results from system simulation, system modeling, and model solving. The results obtained are all in line with expectations. The coherence analysis shows that the quadratic coherence of the random wave-ship swaying system is much greater than the linear coherence, but the linear coherence of the fully developed turbulence is greater for the near Gaussian input type. The reverse computation verification or comparison with real systems indicates that the turbulence simulation and system modeling method in this work have good accuracy and high efficiency in solving algorithms, and the research results can be effectively applied to the model description and system analysis of the quadratic nonlinear systems related to specific turbulent fluctuation excitation response.