With the fierce global competition on satellite networks, the building of satellite constellations grows explosively. Sharply increasing on-orbit data will face the challenge of satellite-ground data transmission. GEO satellites become the top choice for satellite data relay due to their stable satellite-ground link. Most existing spectrum resource management for GEO relays is equipment-oriented and benefit priority, which may lead to a waste of spectrum resources. In this paper, we propose a real-time task-oriented resource allocation strategy for GEO relay systems. We model the spectrum allocation problem as a distributed non-cooperative Stackelberg game process. We prove that when both sides of the game pursue the maximization of personal revenue, the system will enter a Nash equilibrium state, whereas spectrum resources are not fully used. Based on the maximization of individual utilities (U-prior) and spectrum utilization (S-prior) methods, we design an adaptive dual-mode pricing mode to maximize the spectrum resources within a certain loss of revenue. The simulation results show that the S-prior and U-prior have better performance than the baseline method and existing optimization methods. Our proposed dual-mode strategy is making more throughputs and has less delay with little loss of utility values than that of individual utility maximization.