Large Language Models exhibit mode collapse, producing homogeneous outputs that fail to explore valid solution spaces. We present QD-LLM, a framework for parameter-efficient neuroevolution that evolves prompt embeddings, compact neural interfaces (~32K parameters) that steer generation in frozen LLMs (70B+ parameters), within a Quality-Diversity (QD) optimization framework. Our contributions: (1) evolved prompt embeddings via gradient-free optimization enabling behavioral steering without model fine-tuning; (2) hybrid behavior characterization combining semantic and explicit features with formal coverage bounds (Theorem 1) under validated near-independence (NMI $= 0.08 \pm 0.02$); (3) co-evolutionary variation operators including targeted behavioral mutation via finite-difference gradient estimation. On HumanEval (164 problems), MBPP, and creative writing benchmarks, QD-LLM achieves 46.4% higher coverage and 41.4% higher QD-Score than QDAIF ($p<0.001$, 30 runs, Vargha-Delaney $A=0.94$). We demonstrate downstream utility: diverse archives improve test generation (34% more edge cases) and fine-tuning data quality (8.3% accuracy gain). We validate across open-source LLMs (Llama-3-70B, Mistral-Large) with full embedding access, establishing prompt embedding evolution as an effective paradigm bridging neuroevolution and modern LLMs.