Large Language Models (LLMs) are revolutionizing the landscape of Generative Artificial Intelligence (GenAI), with innovative LLM-backed solutions emerging rapidly. However, when applied to database technologies, specifically query generation for graph databases and Knowledge Graphs (KGs), LLMs still face significant challenges. While research on LLM-driven query generation for Structured Query Language (SQL) exists, similar systems for graph databases remain underdeveloped. This paper presents a comparative study addressing the challenge of generating Cypher queries a powerful language for interacting with graph databases using open-access LLMs. We rigorously evaluate several LLM agents (OpenAI ChatGPT 4o, Claude Sonnet 3.5, Google Gemini Pro 1.5, and a locally deployed Llama 3.1 8B) using a designed few-shot learning prompt and Retrieval Augmented Generation (RAG) backed by Chain-of-Thoughts (CoT) reasoning. Our empirical analysis of query generation accuracy reveals that Claude Sonnet 3.5 outperforms its counterparts in this specific domain. Further, we highlight promising future research directions to address the identified limitations and advance LLM-driven query generation for graph databases.