In response to the stability issues of wind turbine generators under random loads, this paper employs a three-mass model where the blades, gearbox, and generator rotor are respectively represented as centralized mass blocks. A stochastic stability analysis method for three-mass wind turbine systems considering random excitations is proposed based on the stochastic averaging method (SAM) within the Hamiltonian framework. First, a stochastic dynamic model of the three-mass wind turbine generator is established by incorporating the random variations of external loads. Second, using the quasi-generalized Hamiltonian principle, the system is formulated into a quasi-Hamiltonian form with its corresponding energy function H. The energy diffusion equation is derived via SAM, and explicit expressions of the mean value and regression square-root process are utilized to obtain the probability density function and regional stability probability. Drift and diffusion coefficients are determined through SAM, enabling the derivation of the backward Kolmogorov equation from the Itô equation. This allows for the calculation of the conditional reliability function and the probability density of the first-passage time. Finally, the effects of load fluctuations with different random intensities on system stability are investigated. Numerical analysis and Monte Carlo simulations verify the effectiveness of the quasi-generalized Hamiltonian stochastic averaging method in the random stability analysis of three-mass wind turbine systems.