Demand prediction is essential for effective management of Mobility-on-Demand (MoD) systems, as accurate forecasts enable better resource allocation, reduced wait times, and improved user satisfaction. Beyond that, probabilistic prediction methods that explicitly account for uncertainty are particularly valuable, as it allows decision-makers to assess risk and make robust plans under uncertain operational environments. However, most existing approaches focus on point predictions, which fail to capture the full spectrum of possible future outcomes. For probabilistic prediction, many methods typically rely on strong parametric distributional assumptions that may not accurately reflect the complex real-world environments. Nonparametric methods proposed in the literature, although promising, often suffer from high computational costs and model complexity, limiting their practical applicability. To overcome these challenges, we propose the Spatial-Temporal Graph Convolutional Network Variational Autoencoder (STGCN-VAE), a novel deep learning framework designed for uncertainty-aware probabilistic travel demand prediction in MoD services. The STGCN-VAE effectively captures complex spatial-temporal dependencies and inherent uncertainties in MoD demand data, generating diverse and realistic future demand scenarios and constructing comprehensive demand distributions. Specifically, the proposed framework integrates three key components: a Spatial-Temporal Graph Convolutional Network (STGCN) to learn complex spatial-temporal dependencies, a Variational Autoencoder (VAE) to compress these patterns into a latent space, and a Kernel Density Estimation (KDE) module to accurately construct probabilistic demand distributions and quantify uncertainties. Experiments on four different real-world MoD datasets including both rideshare and bikeshare services across different cities demonstrate that STGCN-VAE consistently outperforms state-of-the-art baselines in both point and probabilistic prediction, highlighting its robustness and broad transferability across service modes and urban contexts.