The growing computational demands of artificial intelligence (AI) in addressing climate change raise significant concerns about inefficiencies and environmental impact, as highlighted by Jevons' paradox. We propose an attention-enhanced quantum physics-informed neural network (AQ-PINN) to tackle these challenges. This approach integrates quantum computing techniques into physics-informed neural networks (PINNs) for climate modeling, aiming to enhance predictive accuracy in fluid dynamics governed by the Navier-Stokes equations while reducing the computational burden and carbon footprint. By harnessing variational quantum multi-head self-attention mechanisms, our AQ-PINN achieves a 51.51% reduction in model parameters compared to classical multi-head self-attention methods while maintaining comparable convergence and loss. It also employs quantum tensor networks to enhance representational capacity, which can lead to more efficient gradient computations and reduced susceptibility to barren plateaus. Our AQ-PINN represents a crucial step towards more sustainable and effective climate modeling solutions.