Traditional fact-checking relies on humans to formulate relevant and targeted fact-checking questions (FCQs), search for evidence, and verify the factuality of claims. While Large Language Models (LLMs) have been commonly used to automate evidence retrieval and factuality verification at scale, their effectiveness for fact-checking is hindered by the absence of FCQ formulation. To bridge this gap, we seek to answer two research questions: (1) Can LLMs generate relevant FCQs? (2) Can LLM-generated FCQs improve multimodal fact-checking? We therefore introduce a framework LRQ-FACT for using LLMs to generate relevant FCQs to facilitate evidence retrieval and enhance fact-checking by probing information across multiple modalities. Through extensive experiments, we verify if LRQ-FACT can generate relevant FCQs of different types and if LRQ-FACT can consistently outperform baseline methods in multimodal fact-checking. Further analysis illustrates how each component in LRQ-FACT works toward improving the fact-checking performance.