The global energy transition underscores the importance of green hydrogen as a key decarbonization vector. However, its economic and environmental performance is challenged by the temporal mismatch in grid signals, where low electricity prices often coincide with high carbon intensity. This conflict complicates the simultaneous minimization of production cost and carbon emissions. To address this, we propose a novel dual-objective energy management strategy for a renewable-powered hydrogen industrial park, integrating wind, solar, a battery storage system (BESS), and an alkaline water electrolyzer (AWE). A multi-objective optimization model is formulated to minimize both total annual cost and carbon emissions, solved using the ϵ-constraint method to generate Pareto-optimal solutions. A comprehensive case study reveals the economic-environmental trade-offs under varying hydrogen production targets. The results demonstrate that moderate production levels offer the greatest flexibility for trade-offs, while extreme targets constrain the Pareto frontier. Furthermore, operational analysis shows distinct strategies, that is, cost-minimization prioritizes low electricity prices, while carbon-minimization aligns operation with periods of low grid carbon intensity. This work provides a critical decision-making framework for operators and policymakers to navigate the cost-carbon nexus in green hydrogen production.