Satellite–terrestrial integrated networks (STINs) are regarded as a promising solution to meeting the demands of global high-speed seamless network access in the future. Software-defined networking and network function virtualization (SDN/NFV) are two complementary technologies that can be used to ensure that the heterogeneous resources in STINs can be easily managed and deployed. Considering the dual mobility of satellites and ubiquitous users, along with the dynamic requirements of user requests and network resource states, it is challenging to maintain service continuity and high QoE performance in STINs. Thus, we investigate the service migration and reconfiguration scheme, which are of great significance to the guarantee of continuous service provisioning. Specifically, this paper proposes a dynamic service reconfiguration method that can support flexible service configurations on integrated networks, including LEO satellites and ground nodes. We first model the migration cost as an extra delay incurred by service migration and reconfiguration and then formulate the selection processes of the location and migration paths of virtual network functions (VNFs) as an integer linear programming (ILP) optimization problem. Then, we propose a fuzzy logic and quantum genetic algorithm (FQGA) to obtain an approximate optimal solution that can accelerate the solving process efficiently with the benefits of the high-performance computing capacity of QGA. The simulation results validate the effectiveness and improved performance of the scheme proposed in this paper.