We propose AQM-LLM, a framework that distills Large Language Models for Active Queue Management in modern networks. Unlike conventional learning-based AQMs that require extensive feature engineering and struggle with dynamic conditions, our approach leverages the contextual reasoning capability of LLMs to enhance the Low Latency, Low Loss, and Scalable Throughput (L4S) architecture with minimal manual intervention. The L4S-LLM design introduces three key components: 1) a state encoder that transforms heterogeneous network telemetry into token embeddings, 2) a specialized L4S-LLM head that produces congestion actions in a single inference step, and 3) a data-driven Low-Rank Adaptation scheme that drastically reduces trainable parameters while preserving accuracy. Our open-source FreeBSD-14 implementation demonstrates improved queue delay stability and higher bandwidth utilization for both DCTCP and UDP Prague traffic. We emphasize that this work demonstrates architectural feasibility through controlled experiments; deployment on router-class hardware will require additional model optimization (e.g., compression, pruning, or quantization) and is left for future work.