The Blood-Oxygen-Level-Dependent (BOLD) signal of resting-state fMRI (rs-fMRI) records the temporal dynamics of intrinsic functional networks in the brain. However, existing deep learning methods applied to rs-fMRI either neglect the functional dependency between different brain regions in a network or discard the information in the temporal dynamics of brain activity. To overcome those shortcomings, we propose to formulate functional connectivity networks within the context of spatio-temporal graphs. We train a spatio-temporal graph convolutional network (ST-GCN) on short sub-sequences of the BOLD time series to model the non-stationary nature of functional connectivity. Simultaneously, the model learns the importance of graph edges within ST-GCN to gain insight into the functional connectivities contributing to the prediction. In analyzing the rs-fMRI of the Human Connectome Project (HCP, N=1,091) and the National Consortium on Alcohol and Neurodevelopment in Adolescence (NCANDA, N=773), ST-GCN is significantly more accurate than common approaches in predicting gender and age based on BOLD signals. Furthermore, the brain regions and functional connections significantly contributing to the predictions of our model are important markers according to the neuroscience literature.