<p>An exact hydrodynamic formulation of the graphene Dirac equation is derived by performing a complete Madelung decomposition of the two-component spinor field governing massless quasiparticles in monolayer graphene. In this case, beginning with the low-energy Dirac dynamics in the vicinity of a Dirac point, the spinor wave function is recast in terms of hydrodynamic variables composed of probability densities, phase fields, and spin-polarization degrees of freedom. Also, the transformation generates a self-consistent system of nonlinear continuity and momentum equations that follows directly from the underlying Dirac structure and preserves the full dynamical content of the theory without approximation. A stochastic generalization of the hydrodynamic representation is subsequently formulated, yielding the corresponding Fokker-Planck equations together with stochastic Hamilton-Jacobi relations that characterize the probabilistic evolution of graphene quasiparticle transport and incorporate fluctuation-driven contributions within a unified dynamical framework. Moreover, an exact inverse reconstruction scheme is further derived, establishing that the original graphene Dirac equation is uniquely recoverable from the hydrodynamic variables and their governing evolution equations, thereby ensuring mathematical equivalence between the spinorial and hydrodynamic descriptions. The formalism is subsequently extended through the definition of a geometric hydrodynamic reconstruction operator, which enables a covariant generalization to curved spacetime manifolds and effective graphene geometries associated with strain-induced and emergent geometric configurations. In this context, the resulting framework establishes an exact correspondence among spinorial, hydrodynamic, stochastic, and geometric representations of graphene Dirac quasiparticles and shows a unified theoretical basis for the investigation of transport phenomena, fluctuation dynamics, and emergent geometric effects in graphene-based systems.</p>