Influenza-like illness (ILI) remains a persistent global health challenge, necessitating accurate forecasting tools for timely public health response. This study systematically benchmarks fine-tuned large language models (LLMs), e.g., Llama2 and GPT2, for influenza surveillance forecasting in data-limited time-series settings. We develop a lightweight fine-tuning framework that adapts pre-trained LLMs using compact embedding and prediction layers and evaluate it on seven weekly aggregated real-world surveillance datasets. Despite sample sizes of only ～523 time points per region and the absence of cloud-based data transfer, fine-tuned LLMs consistently outperform SARIMA, LSTM, PatchTST, CoVTransformer, FEDformer, Time-LLM, and GPT4TS in both accuracy and stability, especially for long-term forecasts across diverse geographic settings. Even in zero-shot settings, pre-trained LLMs capture broad epidemic trends with performance comparable to SARIMA. These findings establish fine-tuned LLMs as efficient and robust forecasting tools suitable for privacy-sensitive, data-scarce public health applications.