Due to the inherent difficulty in direct observation of traffic demand (including generation, attraction, and assignment), the estimation of origin–destination (OD) poses a significant and intricate challenge in the realm of Intelligent Transportation Systems. As the state-of-the-art methods usually focus on a single traffic demand distribution, accurate estimation of OD in the face of diverse traffic demand and road structures remains a formidable task. To this end, this study proposes a novel model, Cyclic Quasi-Dynamic Transformers (CQDFormer), which leverages forward and backward neural networks for effective OD estimation and traffic assignment. The employment of quasi-dynamic assumption and self-attention mechanism enables CQDFormer to capture the diverse and non-linear characteristics inherent in traffic demand. We utilize calibrated simulations to generate traffic count-OD pairwise data. Additionally, we incorporate real prior matrices and traffic count data to mitigate the distributional shift between simulation and the reality. The proposed CQDFormer is examined using Simuation of Urban Mobility (SUMO), on a large-scale downtown area in Haikou, China, comprising 2328 roads and 1171 junctions. It is found that CQDFormer shows satisfied convergence performance, and achieves a reduction of RMSE by 46.98%, MAE by 45.40% and MAPE by 29.76%, in comparison to the state-of-the-art method with the best performance.