Quantization is a powerful tool to improve large language model (LLM) inference efficiency by utilizing more energy-efficient low-precision datapaths and reducing memory footprint. However, accurately quantizing LLM weights and activations to low precision is challenging without degrading model accuracy. We propose fine-grained mixed precision (FGMP) quantization, a post-training mixed-precision quantization hardware-software co-design methodology that maintains accuracy while quantizing the majority of weights and activations to reduced precision. Our work makes the following contributions: 1) We develop a policy that uses the perturbation in each value, weighted by the Fisher information, to select which weight and activation blocks to keep in higher precision. This approach preserves accuracy by identifying which weight and activation blocks need to be retained in higher precision to minimize the perturbation in the model loss. 2) We also propose a sensitivity-weighted clipping approach for fine-grained quantization which helps retain accuracy for blocks that are quantized to low precision. 3) We then propose hardware augmentations to leverage the efficiency benefits of FGMP quantization. Our hardware implementation encompasses i) datapath support for FGMP at block granularity, and ii) a mixed-precision activation quantization unit to assign activation blocks to high or low precision on the fly with minimal runtime and energy overhead. Our design, prototyped using NVFP4 (an FP4 format with microscaling) as the low-precision datatype and FP8 as the high-precision datatype, facilitates efficient FGMP quantization, attaining <1% perplexity degradation on Wikitext-103 for the Llama-2-7B model relative to an all-FP8 baseline design while consuming 14% less energy during inference and requiring 30% less weight memory.