This study investigates the dynamic and asymmetric connectedness between four crude oil benchmarks (Brent, WTI, INE, Murban) and three climate risk indexes (Physical Risk Index, Transition Risk Index, and U.S. Climate Policy Uncertainty Index). Addressing a critical gap in the literature, which often relies on linear models and average connectedness, we employ the quantile-on-quantile connectedness method to capture non-linear, asymmetric, and state-dependent spillovers, particularly under extreme market conditions. Our analysis reveals that climate risk indexes are predominantly net receivers of shocks from oil markets, with connectedness intensifying sharply during periods of market stress, political conflict, or sudden climate events. The findings highlight that systemic risk is significantly elevated at extreme quantiles, demonstrating that linear models may substantially underestimate true systemic risk during critical junctures. Methodologically, this research demonstrates the efficacy of quantile-on-quantile connectedness in revealing tail-risk effects. Empirically, it provides the most comprehensive comparison to date of connectedness across diverse crude oil benchmarks and climate risk indexes. The results offer crucial insights for investors seeking resilient portfolios, and for policymakers and regulators in designing macro-prudential oversight frameworks that recognize the non-linear and state-dependent nature of climate-financial contagion, emphasizing the need for flexible policies and continuous monitoring.