We investigate classical and quantum geometric information flow theories (GIFs and QGIFs) when the geometric flow evolution and field equations for nonholonomic Einstein systems, NES, are derived from Perelman-Lyapunov-type entropic-type functionals. The term NES encodes models when the fundamental physical equations are subjected to nonholonomic (equivalently, nonintegrable, anholonomic) constraints. There are used canonical geometric variables that allow a general decoupling and integration of systems of nonlinear partial differential equations describing GIFs and QGIFs and Ricci soliton-type configurations. Our approach is different from the constructions elaborated for special classes of solutions characterized by area-hypersurface entropy, related holographic, and dual gauge-gravity models involving generalizations of the Bekenstein-Hawking entropy. We formulate the theory of QGIFs which in certain quasi-classical limits encodes GIFs and models with flow evolution of NES. There are computed, respectively, the von Neumann, relative and conditional entropy; mutual information, entanglement, and Renyi entropy. We construct explicit examples of generic off-diagonal exact and parametric solutions describing stationary solitonic gravitational hierarchies and deformations of black hole configurations. Finally, we show how Perelman's thermodynamic values and extensions to QGIF models can be computed for various new classes of exact solutions which cannot be described following the Bekenstein-Hawking approach.