Effective patient-provider communication is crucial in clinical care, directly impacting patient outcomes and quality of life. Traditional evaluation methods, such as human ratings, patient feedback, and provider self-assessments, are often limited by high costs and scalability issues. Although existing natural language processing (NLP) techniques show promise, they struggle with the nuances of clinical communication and require sensitive clinical data for training, reducing their effectiveness in real-world applications. Emerging large language models (LLMs) offer a new approach to assessing complex communication metrics, with the potential to advance the field through integration into passive sensing and just-in-time intervention systems. This study explores LLMs as evaluators of palliative care communication quality, leveraging their linguistic, in-context learning, and reasoning capabilities. Specifically, using simulated scripts crafted and labeled by healthcare professionals, we test proprietary models (e.g., GPT-4) and fine-tune open-source LLMs (e.g., LLaMA2) with a synthetic dataset generated by GPT-4 to evaluate clinical conversations, to identify key metrics such as `understanding' and `empathy'. Our findings demonstrated LLMs' superior performance in evaluating clinical communication, providing actionable feedback with reasoning, and demonstrating the feasibility and practical viability of developing in-house LLMs. This research highlights LLMs' potential to enhance patient-provider interactions and lays the groundwork for downstream steps in developing LLM-empowered clinical health systems.