The explosive growth of bandwidth-hungry Internet applications have led to the rapid development of new generation mobile network technologies that are expected to provide broadband access to the Internet in a pervasive manner. For example, 6G networks are capable of providing high-speed network access by exploiting higher frequency spectrum; high-throughout satellite communication services are also adopted to achieve pervasive coverage in remote and isolated areas. In order to enable seamless access, integrated satellite-terrestrial communication networks (ISTCNs) have emerged as an important research topic, which aim to provide high-speed and pervasive network access by integrating broadband terrestrial mobile networks with satellite communication systems. As the terrestrial networks have begun to use higher frequency bands that overlap with those of the satellite networks, the satellite terminals can potentially access the terrestrial networks in an integrated manner. However, more efficient spectrum management techniques are needed to reduce the congestion and interference caused by spectrum sharing. In this article, we seek to improve satellite-terrestri-al spectrum sharing performance by introducing non-orthogonal multiple access (NOMA) and cognitive radio (CR) in the spectrum access of the ISTCN. Through fusing NOMA and CR, CR-NO-MA allows multiple satellite terminals to access both the idle and busy spectrum simultaneously, which will achieve high-efficiency full spectrum access. Furthermore, some open research and challenges for the spectrum sharing of the ISTCN are discussed.