A key property of adult stem cells is their ability to persist in a quiescent state for prolonged periods of time. The quiescent state is thought to contribute to stem cell resilience by limiting accumulation of DNA replication–associated mutations. Moreover, cellular stress response factors are thought to play a role in maintaining quiescence and stem cell integrity. We utilized muscle stem cells (MuSCs) as a model of quiescent stem cells and find that the replication stress response protein, ATR (Ataxia Telangiectasia and Rad3-Related), is abundant and active in quiescent but not activated MuSCs. Concurrently, MuSCs display punctate RPA (replication protein A) and R-loop foci, both key triggers for ATR activation. To discern the role of ATR in MuSCs, we generated MuSC-specific ATR conditional knockout (ATRcKO) mice. Surprisingly, ATR ablation results in increased MuSC quiescence exit. Phosphoproteomic analysis of ATRcKO MuSCs reveals enrichment of phosphorylated cyclin F, a key component of the Skp1–Cul1–F-box protein (SCF) ubiquitin ligase complex and regulator of key cell-cycle transition factors, such as the E2F family of transcription factors. Knocking down cyclin F or inhibiting the SCF complex results in E2F1 accumulation and in MuSCs exiting quiescence, similar to ATR-deficient MuSCs. The loss of ATR could be counteracted by inhibiting casein kinase 2 (CK2), the kinase responsible for phosphorylating cyclin F. We propose a model in which MuSCs express cell-cycle progression factors but ATR, in coordination with the cyclin F–SCF complex, represses premature stem cell quiescence exit via ubiquitin–proteasome degradation of these factors.