Brain-computer interface (BCI) has demonstrated its effectiveness in epilepsy treatment and control. In a BCI-aided epilepsy treatment system, therapic electrical stimulus is delivered in response to the prediction of upcoming seizure onsets, therefore timely and accurate seizure prediction algorithm plays an important role. However, unlike typical signatures such as slow or sharp waves in ictal periods, the signal patterns in preictal periods are usually subtle, and highly individual-dependent. How to extract effective and robust preictal features is still a challenging problem. Most recently, graph convolutional neural network (GCNN) has demonstrated the strength in the electroencephalogram (EEG) and intracranial electroencephalogram (iEEG) signal modeling, due to its advantages in describing complex relationships among different EEG/iEEG regions. However, current GCNN models are not suitable for seizure prediction. The effectiveness of GCNNs highly relies on prior graphs that describe the underlying relationships in EEG regions. However, due to the complex mechanism of seizure evolution, the underlying relationship in the preictal period can be diverse in different patients, making it almost impossible to build a proper prior graph in general. To deal with this problem, we propose a novel approach to automatically learn a patient-specific graph in a data-driven way, which is called the joint graph structure and representation learning network (JGRN). JGRN constructs a global-local graph convolutional neural network which jointly learns the graph structures and connection weights in a task-related learning process in iEEG signals, thus the learned graph and feature representations can be optimized toward the objective of seizure prediction. Experimental results show that our JGRN outperforms CNN and GCNN models remarkably, and the improvement is more obvious when preictal features are subtle. The proposed approach promises to achieve robust seizure prediction performance and to have the potential to be extended to general problems in brain-computer interfaces.