Due to the increasing demand for electricity and the inherent uncertainty in power generation, finding efficient solutions to the stochastic alternating current optimal power flow (AC-OPF) problem has become crucial. However, the nonlinear and non-convex nature of AC-OPF, coupled with the growing stochasticity resulting from the integration of renewable energy sources, presents significant challenges in achieving fast and reliable solutions. To address these challenges, this study proposes a novel graph-based generative methodology that effectively captures the uncertainties in power system measurements, enabling the learning of probability distribution functions for generation dispatch and voltage setpoints. Our approach involves modeling the power system as a weighted graph and utilizing a deep spectral graph convolution network to extract powerful spatial patterns from the input graph measurements. A unique variational approach is introduced to identify the most relevant latent features that accurately describe the setpoints of the AC-OPF problem. Additionally, a capsule network with a new greedy dynamic routing algorithm is proposed to precisely decode the latent features and estimate the probabilistic AC-OPF problem. Further, a set of carefully designed physics-informed loss functions is incorporated in the training procedure of the model to ensure adherence to the fundamental physics rules governing power systems. Notably, the proposed physics-informed loss functions not only enhance the accuracy of AC-OPF estimation by effectively regularizing the deep learning model but also significantly reduce the time complexity. Extensive experimental evaluations conducted on various benchmarks demonstrate our proposed model's superiority over both probabilistic and deterministic approaches in terms of relevant criteria.