The national priority of carbon neutrality and decarbonization goals drive the transformation and upgrading of China’s construction industry. The production of green building materials is one of the key ways to decrease carbon emission. To better promote the green development of the construction industry and the realization of the decarbonization goals, a supply-chain system under technology subsidies, inclusive of a general contractor, two green building material manufacturers, is investigated. The Stackelberg model of green building material supply chain is constructed to explore how the green technology innovation of supply chain main body is affected by the technology subsidy. Meanwhile, the backward induction method is utilized to analyze the technology innovation decision-making behavior of green building material players of supply chain under a dynamic game. MATLAB software is used to simulate the stability and dynamic changes of the system in the long-term game. The results show that the decision-making stability of green building material supply chain is beneficial to the stability of supply chain system. The technology subsidy standard with reasonable policy promotes the degree of green building materials to achieve the maximum. However, once the adjustment coefficient of decision-making variables of the main body of the supply chain is large, the supply chain system will fall into chaos and disorder through bifurcation. The delayed feedback control method thus needs to be applied to control chaos, to ensure the sustainable and healthy development of the supply chain of green building materials. This study expands the application scenario of complex system theory in construction supply chain. It also provides theoretical guides for the government to formulate policies to encourage the promotion of green building materials.