The issue of energy consumption in buildings is becoming increasingly pertinent. Society highly values energy conservation and environmental protection, viewing the adoption of green buildings as an inevitable trajectory in the evolution of the construction sector. However, a challenge emerges during the progression of green building construction concerning betrayal within the supply chain. This paper employs a quantum game approach to tackle the issue of green building betrayal within a two-tier green building supply chain comprising developers and contractors. Initially, a non-zero-sum classical game model is formulated based on the green-building construction process between developers and contractors Subsequently, the classical strategy space is extended to the quantum strategy space. The findings indicate that in classical games, and without considering quantum entanglement, even if both parties adopt comprehensive strategies, the party exerting complete effort still faces the risk of betrayal. Conversely, in scenarios of maximum quantum entanglement, the loss incurred by the party not exerting complete effort does not affect the party making a complete effort. Consequently, when accounting for quantum entanglement, both parties tend to opt for complete effort. Lastly, an entanglement treaty is proposed to constrain the strategic choices of both parties and ensure that neither has an incentive to betray. The findings of this study provide good suggestions for green building project managers to improve the effort degree of the developer and general construction contractor in green buildings to promote the development of green buildings.