The green transformation of construction enterprises (GTCEs) is an important way to develop green buildings and realize the goal of “double carbon”. The GTCEs is not only influenced by the internal characteristics of the group but also influenced by the governmental orientation and the pull of the consumer groups. This paper simultaneously considers the heterogeneity of consumer groups and construction enterprise groups, coupling the improved SIR dissemination model, complex network model, and evolutionary game model to describe the dynamic interaction process between construction enterprise groups, government, and consumer groups and to explore the evolution law of GTCEs. The results show that (1) Appropriately increase in green R&D investment by construction enterprises for higher returns, the government’s subsidy and penalty policies and a higher carbon trading price have a positive effect on the GTCEs; (2) a positive social climate, along with the government’s publicity and education, the higher technology level of construction enterprises, and the higher green cognition and lower risk perception level of consumers will strongly promote the GTCEs; and (3) a steady development of the GTCEs is guaranteed by the enterprises’ own inputs and the government’s joint measures on both the supply and demand sides. The conclusions of this study can be used as a reference for the government to formulate policies and for the green transformation and development of construction enterprises.