The applications of Generative Artificial Intelligence (GenAI) and their intersections with data-driven fields, such as healthcare, finance, transportation, and information security, have led to significant improvements in service efficiency and low latency. However, this synergy raises serious concerns regarding the security of large language models (LLMs) and their potential impact on the privacy of companies and users' data. Many technology companies that incorporate LLMs in their services with a certain level of command and control bear a risk of data exposure and secret divulgence caused by insecure LLM pipelines, making them vulnerable to multiple attacks such as data poisoning, prompt injection, and model theft. Although several security techniques (input/output sanitization, decentralized learning, access control management, and encryption) were implemented to reduce this risk, there is still an imminent risk of quantum computing attacks, which are expected to break existing encryption algorithms, hence, retrieving secret keys, encrypted sensitive data, and decrypting encrypted models. In this extensive work, we integrate the Post-Quantum Cryptography (PQC) based Lattice-based Homomorphic Encryption (HE) main functions in the LLM's inference pipeline to secure some of its layers against data privacy attacks. We modify the inference pipeline of the transformer architecture for the LLAMA-3 model while injecting the main homomorphic encryption operations provided by the concrete-ml library. We demonstrate high text generation accuracies (up to 98%) with reasonable latencies (237 ms) on an i9 CPU, reaching up to 80 tokens per second, which proves the feasibility and validity of our work while running a FHE-secured LLAMA-3 inference model. Further experiments and analysis are discussed to justify models' text generation latencies and behaviours.