Recently, Continuous Range Queries have attracted increasing attention from researchers. More data owners are outsourcing their data to cloud servers to minimize storage and computation costs. However, this may introduce potential privacy risks and consume more computational resources. To this end, we propose a Secure Location-based Continuous Range Query scheme in Fog-based Cloud (SCRQ-CF). We first develop a Secure Distributed Multi-level Index (SDMI) structure by combining features of encrypted quadtrees into a two-layer index stored on cloud and fog servers, respectively. Then, we design a Secure Continuous Range Query (SCRQ) processing method in which a novel secure computation protocol is developed to determine the containment relations of index nodes. Next, based on the Cloud-Fog framework, we propose a secure global query processing algorithm in cloud servers and a secure local range query processing algorithm in fog servers. Finally, to improve the efficiency of continuous queries, we propose secure regions for fog servers and incremental updates for result sets in SCRQ processing. Theoretical analysis demonstrates that our scheme achieves security and effectiveness in supporting SCRQ. Experimental evaluations on four datasets confirm its superior performance over existing schemes.