Generalized Disjunctive Programming (GDP) is a framework for optimization problems involving discrete decisions and nonlinear constraints. The widely used Hull Reformulation transforms GDPs into Mixed-Integer Nonlinear Programming (MINLP) problems with tighter continuous relaxations. However, it typically employs an epsilon approximation of the closure of the perspective function for nonlinear constraints, which can lead to numerical instability and a weaker continuous relaxation. We present an exact Hull Reformulation for GDPs with quadratic constraints, preserving the original quadratic structure and avoiding relaxation weakening. The method applies to both convex and non-convex constraints and extends earlier approaches developed for second-order cone representable functions. We prove equivalence to the conventional exact Hull Reformulation for quadratic constraints and demonstrate improved computational performance and numerical stability in extensive computational experiments. Benchmarks include random GDP instances, Continuously Stirred Tank Reactor network optimization, k-means clustering, and constrained layout optimization problems. The computational results demonstrate improved solution times and numerical stability of the proposed exact Hull Reformulation, making it a preferable approach for GDP problems with quadratic constraints.