Origin-Destination (O-D) travel demand prediction is a fundamental challenge in transportation. Recently, spatial-temporal deep learning models demonstrate the tremendous potential to enhance prediction accuracy. However, few studies tackled the uncertainty and sparsity issues in fine-grained O-D matrices. This presents a serious problem, because a vast number of zeros deviate from the Gaussian assumption underlying the deterministic deep learning models. To address this issue, we design a Spatial-Temporal Zero-Inflated Negative Binomial Graph Neural Network (STZINB-GNN) to quantify the uncertainty of the sparse travel demand. It analyzes spatial and temporal correlations using diffusion and temporal convolution networks, which are then fused to parameterize the probabilistic distributions of travel demand. The STZINB-GNN is examined using two real-world datasets with various spatial and temporal resolutions. The results demonstrate the superiority of STZINB-GNN over benchmark models, especially under high spatial-temporal resolutions, because of its high accuracy, tight confidence intervals, and interpretable parameters. The sparsity parameter of the STZINB-GNN has physical interpretation for various transportation applications.