It is a common practice in natural language processing to pre-train a single model on a general domain and then fine-tune it for downstream tasks. However, when it comes to Large Language Models, fine-tuning the entire model can be computationally expensive, resulting in very intensive energy consumption. As a result, several Parameter efficient fine-tuning (PEFT) approaches were recently proposed. One of the most popular approaches is low-rank adaptation (LoRA), where the key insight is decomposing the update weights of the pre-trained model into two low-rank matrices. However, the proposed approaches either use the same rank value across all different weight matrices, or do not use any quantization technique, which has been shown to be one of the most important factors when it comes to a model's energy consumption. In this work, we propose Bayesian-LoRA (B-LoRA) which approaches matrix decomposition and quantization from a Bayesian perspective by employing a prior distribution on both quantization levels and rank values of the learned low-rank matrices. As a result, B-LoRA is able to fine-tune a pre-trained model on a specific downstream task, finding the optimal rank values and quantization levels for every low-rank matrix. We validate the proposed model fine-tuning a pre-trained DeBERTaV3 on the GLUE benchmark. Moreover, we compare it to relevant baselines and present both qualitative and quantitative results, showing how B-LoRA is able to learn optimal-rank quantized matrices, performs on par or better than baselines while reducing the total amount of bit operations of roughly 70\% with respect to the baselines ones.