Underwater glider (UG) plays an important role in ocean observation and exploration for a more efficient and deeper understanding of complex ocean environment. Timely identifying the motion states of UG is conducive for timely attitude adjustment and detection of potential anomalies, thereby improving the working reliability of UG. Combining limited penetrable visibility graph (LPVG) and graph convolutional networks (GCN) with self-attention mechanisms, we propose a novel method for motion states identification of UG, which is called as visibility graph and self-attention mechanism-based graph convolutional network (VGSA-GCN). Based on the actual sea trial data of UG, we chose the attitude angle signals of motion states related sensors collected by the control system of UG as the research object and constructed complex networks based on the LPVG method from pitch angle, roll angle, and heading angle data in diving and climbing states. Then, we build a self-attention mechanism-based GCN framework and classify the graphs under different motion states constructed by a complex network. Compared with support vector machines, convolutional neural network, and GCN without self-attention pooling layer, the proposed VGSA-GCN method can more accurately distinguish the diving and climbing states of UG. Subsequently, we analyze the variation of the transitivity coefficient corresponding to these two motion states. The results suggest that the coordination of the various sensors in the attitude adjustment unit during diving becomes closer and more efficient, which corresponds to the higher network measure of the diving state compared to the climbing state.