We present LinkQ, a system that leverages a large language model (LLM) to facilitate knowledge graph (KG) query construction through natural language question-answering. Traditional approaches often require detailed knowledge of complex graph querying languages, limiting the ability for users -- even experts -- to acquire valuable insights from KG data. LinkQ simplifies this process by first interpreting a user's question, then converting it into a well-formed KG query. By using the LLM to construct a query instead of directly answering the user's question, LinkQ guards against the LLM hallucinating or generating false, erroneous information. By integrating an LLM into LinkQ, users are able to conduct both exploratory and confirmatory data analysis, with the LLM helping to iteratively refine open-ended questions into precise ones. To demonstrate the efficacy of LinkQ, we conducted a qualitative study with five KG practitioners and distill their feedback. Our results indicate that practitioners find LinkQ effective for KG question-answering, and desire future LLM-assisted systems for the exploratory analysis of graph databases.