Fiber-reinforced composites and biological soft tissues exhibit nonlinear viscoelastic and directionally dependent behaviors under finite deformations. The proposed nonlinear anisotropic viscoelastic modeling framework integrates three basic characteristics: (1) the Green-Naghdi kinematic assumption being employed as the decoupled mechanism between elastic and inelastic deformations [1], (2) the directional information being encoded into strain energy and the dissipation potential through the rank-four structural tensors, (3) separate dissipation potential being specified for individual components of the material. The proposed Green-Naghdi kinematic decomposition employs internal variables, analogous to the generalized strain tensors, to describe the irreversible (non-equilibrium) deformations of the materials. Compared with the multiplicative decomposition of the deformation gradient, the advantage of adopting the Green-Naghdi kinematic assumption lies in the unnecessity of the intermediate configuration. Thus, for anisotropic materials, the consideration of whether the direction and distribution of fibers need to remain consistent in the reference configuration and intermediate configuration becomes unnecessary. Leveraging the rank-four structural tensors, where the directional information is retained, we express the strain energy and the dissipation potential as quadratic forms of generalized strain tensors and internal variable rates [2], respectively. The proposed framework specifies independent, non-negative, and convex dissipation potential for respective constituents of the composite materials. For individual components of the material, the internal variable must satisfy the maximum dissipation principle and guarantee thermodynamic consistency with the Clausius-Planck inequality, leading to the viscous evolution equation. In this presentation, we validate the stretch inversion phenomenon caused by the instability, arising from the constraint on the end of the fiber, as well as the elliptically shaped hysteresis caused by the viscous evolution of circumferential stretch. Regarding the above properties, we present an efficient algorithm with finite-element implementation in a purely Lagrangian configuration, gaining the capability to trace the equilibrium path of the nonlinear dynamical response.