Indoor localization is a critical enabler for a wide range of location-based services in smart environments, including navigation, asset tracking, and safety-critical applications. Recent graph-based models leverage spatial relationships between Wire-less Fidelity (Wi-Fi) Access Points (APs) and devices, offering finer localization granularity, but fall short in quantifying prediction uncertainty, a key requirement for real-world deployment. In this paper, we propose Spatially-Adaptive Conformal Graph Transformer (SAC-GT), a framework for accurate and reliable indoor localization. SAC-GT integrates a Graph Transformer (GT) model that captures network's spatial topology and signal strength dynamics, with a novel Spatially-Adaptive Conformal Prediction (SACP) method that provides region-specific uncertainty estimates. This allows SAC-GT to produce not only precise two-dimensional (2D) location predictions but also statistically valid confidence regions tailored to varying environmental conditions. Extensive evaluations on a large-scale real-world dataset demonstrate that the proposed SAC-GT solution achieves state-of-the-art localization accuracy while delivering robust and spatially adaptive reliability guarantees.