Aberrant expressions of long non-coding RNAs (lncRNAs) are often associated with diseases and identification of disease-related lncRNAs is helpful for elucidating complex pathogenesis. Recent methods for predicting associations between lncRNAs and diseases integrate their pertinent heterogeneous data. However, they failed to deeply integrate topological information of heterogeneous network comprising lncRNAs, diseases, and miRNAs. We proposed a novel method based on the graph convolutional network and convolutional neural network, referred to as GCNLDA, to infer disease-related lncRNA candidates. The heterogeneous network containing the lncRNA, disease, and miRNA nodes, is constructed firstly. The embedding matrix of a lncRNA-disease node pair was constructed according to various biological premises about lncRNAs, diseases, and miRNAs. A new framework based on a graph convolutional network and a convolutional neural network was developed to learn network and local representations of the lncRNA-disease pair. On the left side of the framework, the autoencoder based on graph convolution deeply integrated topological information within the heterogeneous lncRNA-disease-miRNA network. Moreover, as different node features have discriminative contributions to the association prediction, an attention mechanism at node feature level is constructed. The left side learnt the network representation of the lncRNA-disease pair. The convolutional neural networks on the right side of the framework learnt the local representation of the lncRNA-disease pair by focusing on the similarities, associations, and interactions that are only related to the pair. Compared to several state-of-the-art prediction methods, GCNLDA had superior performance. Case studies on stomach cancer, osteosarcoma, and lung cancer confirmed that GCNLDA effectively discovers the potential lncRNA-disease associations.