As global climate change and environmental issues escalate, carbon reduction has emerged as a paramount global concern. Agriculture accounts for approximately 30% of global greenhouse gas emissions, making carbon reduction in this sector crucial for attaining global emission targets. Carbon sink trading serves as a supplementary mechanism to achieve carbon peaking and neutrality, helping to lower the rate of carbon emissions. However, practical projects and research in the field of carbon sink trading are not enough currently. This work aims to thoroughly explore the cooperative models between farmers and retailers within the context of agricultural carbon sink trading, as well as the optimal decisions on the efforts to reduce carbon emission for both parties under different cooperative models. To this end, we delve into three distinct cooperative frameworks: the decentralized, the Stackelberg, and the centralized models, each accompanied by a corresponding differential game model. The Hamilton-Jacobi-Bellman equation is utilized to investigate the equilibrium strategies of each participant under these three cooperative models, respectively. Furthermore, we conducted numerical simulations to analyze the carbon emission reduction efforts of farmers and retailers, the carbon emission reduction level of the agricultural supply chain, and the overall profits of the supply chain. We also compare scenarios with and without carbon sink trading to provide a comprehensive assessment. The numerical results indicate that the centralized model excels in all aspects, followed by the Stackelberg model, with the decentralized model showing the weakest performance. Additionally, carbon sink trading can significantly increase the profits of the participants under each cooperative model.