This paper proposes finite-time (FT) and predefined-time (PdT) neurodynamic models with time-varying coefficients for solving inverse quasi-variational inequality (IQVI) problems. Two projection-based neurodynamic models are developed, in which time-varying gains are designed to regulate the convergence speed and improve transient performance. A nominal projected model is introduced to establish the equivalence between its equilibrium points and the solutions of IQVI problems. Under Lipschitz continuity and strong monotonicity assumptions, the existence, uniqueness, and global convergence of the proposed models are established. By employing Lyapunov theory, FT and PdT convergence are rigorously proved together with explicit settling-time estimates. In particular, the PdT model guarantees convergence within a user-prescribed time bound independent of the initial conditions. Furthermore, a forward Euler discretization is developed to obtain implementable numerical algorithms, and robustness of the PdT model under bounded disturbances is established. Numerical experiments, including an additional higher-dimensional IQVI problem example and a sparse signal recovery application, validate the theoretical results and demonstrate the effectiveness, robustness, and computational efficiency of the proposed framework.