Information Geometry (IG) offers a differential geometric perspective for examining statistical models, where families of probability distributions are regarded as manifolds. This paper presents a new method for modeling the IG of a stable M/G/1 queueing system. By defining the M/G/1 queue manifold, the Fisher Information Matrix (FIM) and its inverse (IFIM) are derived, demonstrating the critical influence of system stability on the existence of IFIM. Divergence measures like Kullback's Divergence (KD) and J-Divergence (JD) are explored, revealing the effects of server utilization and the squared coefficient of variation on these measures. It is established that the stable M/G/1 queue manifold is developable (with zero Gaussian curvature) while containing a non-zero Ricci Curvature Tensor (RCT), along with new dynamics that connect RCT behavior to different stability phases. The Information Matrix Exponential (IME) is introduced, illustrating its instability and its reverse correlation with the stability of the M/G/1 manifold. The investigation is broadened to include other divergence measures, merging Queueing Theory, IG, and Riemannian Geometry. Connections to advanced machine learning and metric learning are examined, emphasizing queue learning as an innovative approach. A comprehensive analytical exploration incorporates Gaussian and Ricci curvatures, the Einstein Tensor, and the Stress-Energy Tensor, offering insights into their stability dynamics and geometric significance. For the first time, this research provides thorough derivations of Gaussian curvature, RCT, and their relationships with stability analysis. Computational IG is employed to visualize queueing systems, forging new links between Queueing Theory, matrix theory, IG, and the Theory of Relativity. This cohesive framework enhances both theoretical and practical understandings of queueing systems.