Considering the supply chain composed of a power battery supplier and a new energy vehicle manufacturer, under the carbon cap-and-trade policy, this paper studies the different cooperation modes between the manufacturer and the supplier as well as their strategies for green technology and power battery production. Three game models are constructed and solved, respectively, under the collaboration mode of wholesale purchasing, patent-licensed manufacturing, and own R&D + Wholesale purchasing. The equilibrium analysis is carried out. Finally, the influence of relevant parameters is explored through numerical simulation. It is found that (1) the manufacturer’s choice of optimal battery production strategy is influenced by the input cost of green technology, the production cost of power battery, the carbon trading price, and the free carbon quota allocated by the government; (2) the cost coefficient of technological innovation affects negatively the optimal decision-making of the supply chain members, the market demand, and the optimal profit, and it has no impact when the cost coefficient reaches a certain value; (3) carbon cap-and-trade policy can, to a certain extent, incentivize suppliers and manufacturers to carry out technological innovation to reduce carbon emissions in the production process, but we cannot ignore the negative impacts of excessively high carbon trading price on the level of emission reduction and the market demand; and (4) the government should reasonably control the carbon price and carbon quota. The above conclusion will provide reference suggestions for new energy vehicle manufacturers and related suppliers.