This paper examines the dynamics of carbon quota pricing and emissions under government subsidies in the EU carbon market. The study reveals three core findings. First, carbon emissions, carbon subsidies, and carbon quota prices exhibit strong interconnectedness with pronounced seasonality, and spillover effects intensify during exogenous shocks, particularly at short-term frequencies within one to five days. Second, Germany consistently serves as the primary source of spillover effects, reflecting its energy-intensive industrial structure and dominant position in the EU Emissions Trading System, while carbon subsidies function as information receivers in the short term but maintain stable roles in the medium to long term under the EU's sustained climate commitments. Third, the COVID-19 pandemic fundamentally altered spillover patterns, with the EU industrial sector transitioning from a net transmitter to a net recipient of spillovers and carbon quota prices shifting to net recipients during the shock. These findings are derived from a stochastic differential game model that captures strategic interactions between governments and enterprises under exogenous shocks, combined with TVP-VAR spillover analysis that quantifies dynamic connectedness across time and frequency domains. The results enhance the understanding of carbon market mechanisms under policy interventions and external disturbances, offering insights for the development of more efficient and resilient carbon trading systems.