This paper investigates a fractional-order human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) model with generalized nonlinear incidence rates $f(S,I)$ and $g(S,E)$. First, the existence, uniqueness, non-negativity, and boundedness of the model solutions are proven, and the basic reproduction number $R_0^{\alpha }$ is derived. The analysis indicates that the model exhibits two equilibrium points: A disease-free equilibrium and an endemic equilibrium. The local asymptotic stability of each equilibrium point is examined using the Routh-Hurwitz criterion. Additionally, by employing Lyapunov functionals and applying LaSalle's invariance principle, the global stability of the equilibrium points is demonstrated. The main conclusion is that under appropriate conditions, if $R_0^{\alpha }<1$, then the disease will eventually disappear, whereas if $R_0^{\alpha }>1$, then it will persist. Finally, the model is utilized to predict and control of HIV/AIDS transmission in Mexico, thereby highlighting the role of mutural preventive measures adopted by susceptible individuals and HIV-infected individuals in reducing disease spread. Simulations are performed to confirm the theoretical validity and practical significance of the model.