In cross-device Federated Learning (FL), the communication cost of transmitting full-precision models between edge devices and a central server is a significant bottleneck, due to expensive, unreliable, and low-bandwidth wireless connections. As a solution, we propose a novel FL framework named QSFL, towards optimizing FL uplink (client-to-server) communication at both client and model levels. At the client level, we design a Qualification Judgment (QJ) algorithm to sample high-qualification clients to upload models. At the model level, we explore a Sparse Cyclic Sliding Segment (SCSS) algorithm to further compress transmitted models. We prove that QSFL can converge over wall-to-wall time, and develop an optimal hyperparameter searching algorithm based on theoretical analysis to enable QSFL to make the best trade-off between model accuracy and communication cost. Experimental results show that QSFL achieves state-of-the-art compression ratios with marginal model accuracy degradation.