Post-quantum cryptography (PQC) is crucial for securing data against emerging quantum threats. However, its algorithms are computationally complex and difficult to implement efficiently on hardware. In this paper, we explore the potential of Large Language Models (LLMs) to accelerate the hardware-software co-design process for PQC, with a focus on the FALCON digital signature scheme. We present a novel framework that leverages LLMs to analyze PQC algorithms, identify performance-critical components, and generate candidate hardware descriptions for FPGA implementation. We present the first quantitative comparison between LLM-guided synthesis and conventional HLS-based approaches for low-level compute-intensive kernels in FALCON, showing that human-in-the-loop LLM-generated accelerators can achieve up to 2.5x speedup in kernel execution time with significantly fewer cycles, while highlighting trade-offs in resource utilization and power consumption. Our results suggest that LLMs can minimize design effort and development time by automating FPGA accelerator design iterations for PQC algorithms, offering a promising new direction for rapid and adaptive PQC accelerator design on FPGAs.