Graph Convolutional Networks (GCNs) have emerged as the state-of-the-art deep learning model for representation learning on graphs. However, it remains notoriously challenging to train and inference GCNs over large graph datasets, limiting their application to large real-world graphs and hindering the exploration of deeper and more sophisticated GCN graphs. This is because as the graph size grows, the sheer number of node features and the large adjacency matrix can easily explode the required memory and data movements. To tackle the aforementioned challenges, we explore the possibility of drawing lottery tickets when sparsifying GCN graphs, i.e., subgraphs that largely shrink the adjacency matrix yet are capable of achieving accuracy comparable to or even better than their full graphs. Specifically, we for the first time discover the existence of graph early-bird (GEB) tickets that emerge at the very early stage when sparsifying GCN graphs, and propose a simple yet effective detector to automatically identify the emergence of such GEB tickets. Furthermore, we advocate graph-model co-optimization and develop a generic efficient GCN early-bird training framework dubbed GEBT that can significantly boost the efficiency of GCN training by (1) drawing joint early-bird tickets between the GCN graphs and models and (2) enabling simultaneously sparsification of both the GCN graphs and models. Experiments on various GCN models and datasets consistently validate our GEB finding and the effectiveness of our GEBT, e.g., our GEBT achieves up to 80.2% ~ 85.6% and 84.6% ~ 87.5% savings of GCN training and inference costs while offering a comparable or even better accuracy as compared to state-of-the-art methods. Our source code and supplementary appendix are available at https://github.com/RICE-EIC/Early-Bird-GCN.