Based on the framework of differential inclusion, we design a quaternion-valued neural network (QVNN) to solve a class of nonsmooth nonconvex optimization problems with equality, bounded and inequality constraints in quaternion domain. The modeling of the network avoids the calculation of penalty factors and guarantees the convergence in finite time to the feasible region and the sets of critical points of the nonconvex optimization problems. The global existence of the state solution of the network can be obtained by nonsmooth analysis. Our approach is directly used to solve nonconvex optimization problems over quaternion domain without splitting them into real or complex domain, and the theoretical results about convergence are also completely established in quaternion domain. Additionally, the feasibility and effectiveness of the proposed quaternion-valued neural network approach are demonstrated by numerical experiments and the application of attitude estimation for micro quadrotor.