This paper presents a new complete first-order temporal BDI logic and forest multi-agent system. The main characteristic of the logic is that its semantic model is based on the forest multi-agent system, which enables us to reason about beliefs, desires, and intentions between agents with different layers such as father agent and child agent. The logical reasoning and hierarchical structures of the forest multi-agent system can suitably capture the hierarchical property of the real systems and therefore is practically realistic. We propose further four classes of first-order BDI interpreted systems and four proof systems which are sound and complete with respect to corresponding classes of BDI interpreted systems. Finally, we give a case to show how to characterize the forest multi-agent system by using the hierarchical structure of modules, and to solve the model checking problem of first-order temporal BDI logic for the forest multi-agent system.