As escalating environmental pollution results from economic development, the green supply chain is vital in enhancing ecological quality. In light of the rapid growth of online shopping, the dual-channel supply chain has gained significant popularity. Governments have implemented policies to achieve carbon peaks and neutrality worldwide. Considering green quality and channel preferences, this study constructs a Stackelberg model led by the manufacturer within a dual-channel supply chain to compare the effects under two government subsidy strategies. A Stackelberg game-theoretic approach is applied to analyze the model. The findings indicate the following: (1) Increases in the subsidy coefficient can increase product greenness and the overall profit in both subsidy strategies, which makes sense; (2) When direct sales channels become dominant, subsidizing the manufacturer proves superior to subsidizing the retailer in terms of promoting green quality and overall profit, which extends existing studies of government subsidy decisions. Furthermore, subsidizing the retailer may negatively affect the total supply chain profit; (3) Consistent with previous literature, intense competition between manufacturers and retailers is expected to enhance the overall profit of the supply chain; (4) Interestingly, customer preference for online channels does not influence product green quality under manufacturer subsidization strategies but exhibits a negative impact under retailer subsidization strategies. Finally, this research provides managerial implications for decision-makers and potential issues for future research.