The advancement of 6G technology significantly enhances the Internet of Things (IoT) applications, especially in remote areas where traditional cellular infrastructure is not feasible. Satellite communication, a crucial component of 6G, extends IoT connectivity to these underserved regions. In this context, the growing interest in low earth orbit (LEO) satellite communication stems from its recent advancements in offering high data rate services and minimizing service latency. Next-generation LEO satellite systems, with regenerative capabilities, allow for adaptability in bandwidth management and on-board data processing. However, the scarcity of satellite spectrum presents a barrier to the expansion of LEO satellite networks and the development of integrated terrestrial-space infrastructures. To address this challenge, we propose constructing a Radio Environment Map (REM) aboard LEO satellites to opportunistically tap into the unused spectrum of geostationary (GEO) satellites within a cognitive GEO-LEO satellite IoT network. This solution facilitates REM construction through collaboration among neighboring LEO satellites while also considering the frequency reuse scheme of GEO satellites. Our REM construction approach leverages cyclostationary-based sensing at LEO satellites, serving the dual purpose of REM construction and Doppler shift estimation to track multiple GEO frequency signals. Following REM construction, LEO satellites utilize deep learning techniques to predict GEO spectrum occupancy without further sensing, thereby optimizing secondary spectrum utilization of the IoT network. We propose a deep learning neural network architecture based on a sequence-to-sequence model tailored for spectrum prediction at LEO satellites. Simulations demonstrate superior performance in detection probability of the proposed deep learning network compared to convolutional long short-term memory networks, achieving this with lower computational complexity.