Query processing in road networks has been studied extensively in recent years. However, the processing of moving queries in road networks has received little attention. In this paper, we introduce a new algorithm called the Safe Exit Algorithm (SEA), which can efficiently compute the safe exit points of a moving nearest neighbor (NN) query on road networks. The safe region of a query is an area where the query result remains unchanged, provided that the query remains inside the safe region At each safe exit point, the safe region of a query and its non-safe region meet so that a set of safe exit points represents the border of the safe region. Before reaching a safe exit point, the client (query object) does not have to request the server to re-evaluate the query This significantly reduces the server processing costs and the communication costs between the server and moving clients. Extensive experimental results show that SEA outperforms a conventional algorithm by up to two orders of magnitude in terms of communication costs and computation costs.