In digital healthcare, large language models (LLMs) have primarily been utilized to enhance question-answering capabilities and improve patient interactions. However, effective patient care necessitates LLM chains that can actively gather information by posing relevant questions. This paper presents HealthQ, a novel framework designed to evaluate the questioning capabilities of LLM healthcare chains. We implemented several LLM chains, including Retrieval-Augmented Generation (RAG), Chain of Thought (CoT), and reflective chains, and introduced an LLM judge to assess the relevance and informativeness of the generated questions. To validate HealthQ, we employed traditional Natural Language Processing (NLP) metrics such as Recall-Oriented Understudy for Gisting Evaluation (ROUGE) and Named Entity Recognition (NER)-based set comparison, and constructed two custom datasets from public medical note datasets, ChatDoctor and MTS-Dialog. Our contributions are threefold: we provide the first comprehensive study on the questioning capabilities of LLMs in healthcare conversations, develop a novel dataset generation pipeline, and propose a detailed evaluation methodology.