This study develops a novel (q, τ )-quantum deformation framework for fixed point theory and its applications to fractional differential equations and optimization. By introducing a deformation dependent control function and the associated (q, τ )-metric structure, we establish a Banach type fixed point theorem with explicit convergence estimates governed by the (q, τ )-Gamma function. The obtained framework rigorously characterize the metricity and completeness of the deformed space and provide quantitative bounds on the rate of convergence of Picard iterations in terms of the deformation parameters (q, τ ). The developed theory is applied to nonlinear fractional problems driven by the (q, τ )-fractional operator Dα q,τ . Under explicit deformation dependent conditions, we prove existence and uniqueness of solutions and derive stability estimates that reveal how the (q, τ )-Gamma normalization regulates memory intensity and convergence speed. In addition, we formulate and analyze optimal control problems for (q, τ )-fractional systems. A complete first order optimality system is derived, including adjoint equations and projection characterizations of the optimal control, and the well posedness of the state and adjoint problems is established via the proposed fixed point framework. Numerical experiments validate the analytical results and demonstrate the effectiveness of the deformation parameters in tuning convergence behavior for both dynamical and optimization problems. In particular, the simulations show monotone decay of Picard iterates and objective functionals, confirming the stabilizing role of the (q, τ )-Gamma function. The proposed framework thus provides a unified and robust analytical and computational foundation for the study and optimization of quantum deformed fractional systems.