The complex risks of global climate change and the transition to a sustainable economy have increasingly become central to research and policy debates. Climate risk perceptions influence fossil fuel, renewable energy, and carbon markets through both investment behavior and regulatory policy channels. Understanding the spillovers between climate risk perceptions and commodity markets has profound implications for sustainable investments and risk management strategies. This paper extends the existing literature by examining higher-order moment risk spillovers among perceptions of climate physical risks (CPR) and transition risks (CTR), fossil fuel, renewable energy, and carbon markets across different quantiles. Furthermore, this paper also proposes an analytical framework that integrates ex-post moment measures with an innovative QVAR extended joint connectedness approach. Our empirical analysis reveals that the connectedness outcomes are contingent upon moment orders and specific quantile levels. Notably, total spillovers are markedly higher at the extreme quantiles (especially at the 0.95 quantile) compared to the median quantile. Importantly, CPRI and CTRI serve as net transmitters of spillovers at the 0.05 and 0.95 quantiles but shift to being net recipients under normal market conditions. The directional net spillovers transmitted from climate risk perceptions to energy and carbon markets are more pronounced and consistent at the extreme higher and lower quantiles. Finally, we find that dynamic total spillovers of skewness and kurtosis at extreme quantiles are more volatile than at the median, with significant sensitivity to major events such as the COVID-19 pandemic, the Russia-Ukraine war, the Israel-Hamas war, extreme climate disasters, and the United Nations Climate Change Conferences.