This article, for the first time, extends the zeroing neural network (ZNN) method to address the problem of dynamic quaternion-valued matrix inversion. Due to the noncommutative property of quaternion multiplication, the complex representation method is first adopted to transform quaternion-valued matrices into the corresponding complex-valued matrices. Then, based on two kinds of ways to deal with nonlinear activation functions in the complex-valued domain, this article proposes two quaternion-valued ZNN (QVZNN) models for dynamic quaternion-valued matrix inversion. In addition, a novel nonlinear activation function is given to accelerate the convergence rate of the models to reach the predefined-time convergence. The detailed theoretical analysis, together with four theorems, are given to show the excellent properties of the QVZNN models. Furthermore, the upper bound of the convergence time is derived analytically with the residual error being zero theoretically. Finally, two numerical examples are provided to verify the theoretical results and the effectiveness of the QVZNN models for the dynamic quaternion-valued matrix inversion, and an application to mobile manipulator control is provided to indicate the practical application value of the QVZNN models.