Identification of nonlinear dynamic systems is a critical task in various fields. Artificial neural networks have been widely used for this purpose due to their ability to approximate complex functions. However, their computational efficiency and stability often pose challenges, especially in real-time applications. Quantum computation has shown potential for enhancing computational performance, but its integration with neural networks is still under investigation. The primary motivation addressed in this paper is the development of an effective strategy for synthesizing and applying recurrent quantum neural networks based on Lyapunov stability criteria (RQNN-LS) for nonlinear system identification. This model enhances the computational efficiency of recurrent neural networks by incorporating quantum computation into the neural network characteristics by using qubit neurons for data processing. Additionally, adaptive learning rates are derived based on Lyapunov stability theory for online tuning of the parameters to guarantee the stability of the proposed technique. The applicability and superiority of the presented RQNN-LS identifier are verified through the simulation and practical results of nonlinear system identification, comparing its performance with other existing identification techniques. The comparative results demonstrated significant improvements in computational efficiency with the proposed technique and highlighted the merits and superiority of the developed model over other methodologies.