Large Language Model (LLM) serving systems increasingly face strict time-to-first-token (TTFT) service-level objectives (SLOs), yet TTFT remains highly sensitive to router-side queueing effects. Prefill costs scale with prompt length, decode lengths are uncertain, and prefix locality creates strong performance skew across requests. Despite major advances in continuous batching and KV-cache management, today’s routers are often agnostic to request cost, which makes them vulnerable to head-of-line blocking and tail-latency amplification under mixed workloads. We propose QUARTZ, a quantile-aware routing and queueing layer for LLM serving that predicts conservative quantile-based request-cost proxies, rather than point estimates, using lightweight router-visible signals. QUARTZ uses these quantiles together with backlog-aware router signals to guide worker selection and admission decisions that better align with TTFT tail SLOs while preserving fairness. We implement QUARTZ as a router upgrade for SGLang and evaluate it on representative interactive and retrieval-augmented workloads. The results show reductions in TTFT tail latency and SLO violations across heterogeneous workloads.