Space-ground integrated network, a strategic, driving, and irreplaceable infrastructure, guarantees the development of economic and national security. However, its natures of limited resources, frequent handovers, and intermittently connected links significantly reduce the quality of service. To address this issue, a quality-of-service-aware dynamic evolution model is proposed based on complex network theory. On one hand, a quality-of-service-aware strategy is adopted in the model. During evolution phases of growth and handovers, links are established or deleted according to the qualityof-service-aware preferential attachment following the rule of better quality of service getting richer and worse quality of service getting poor or to die. On the other hand, dynamic handover of nodes and intermittent connection of links are taken into account and introduced into the model. Meanwhile, node heterogeneity is analyzed and heterogeneous nodes are endowed with discriminate interactions. Theoretical analysis and simulations are utilized to explore the degree distribution and its characteristics. Results reveal that this model is a scale-free model with drift power-law distribution, fat-tail and small-world effect, and drift character of degree distribution results from dynamic handover. Furthermore, this model exerts well fault tolerance and attack resistance compared to signal-strength-based strategy. In addition, node heterogeneity and quality-of-service-aware strategy improve the attack resistance and overall quality of service of space-ground integrated network.