In response to consumer environmental awareness and carbon taxation, we develop a nonlinear Cournot duopoly model to examine how firms make green reputation decisions. Using symbolic computation methods, we analyze equilibrium strategies in both static and dynamic settings. Our results show that a firm's emission abatement effort increases with its basic production cost, green efficiency, and carbon tax rate. Under nonlinear demand, less productive firms may exert greater effort to improve their green reputation, a phenomenon not observed under linear demand. Numerical simulations also show that excessive carbon taxes or low green efficiency can disrupt market equilibrium. Our study offers policymakers and business practitioners practical insights into how regulatory incentives and consumer preferences shape firms' green strategies. The proposed framework supports the development of environmental policies that foster green transformation and support long-term market stability.