The SPARQL query language is the standard method to access knowledge graphs (KGs). However, formulating SPARQL queries is a significant challenge for non-expert users, and remains time-consuming for the experienced ones. Best practices recommend to document KGs with competency questions and example queries to contextualise the knowledge they contain and illustrate their potential applications. In practice, however, this is either not the case or the examples are provided in limited numbers. Large Language Models (LLMs) are being used in conversational agents and are proving to be an attractive solution with a wide range of applications, from simple question-answering about common knowledge to generating code in a targeted programming language. However, training and testing these models to produce high quality SPARQL queries from natural language questions requires substantial datasets of question-query pairs. In this paper, we present Q${}^2$Forge that addresses the challenge of generating new competency questions for a KG and corresponding SPARQL queries. It iteratively validates those queries with human feedback and LLM as a judge. Q${}^2$Forge is open source, generic, extensible and modular, meaning that the different modules of the application (CQ generation, query generation and query refinement) can be used separately, as an integrated pipeline, or replaced by alternative services. The result is a complete pipeline from competency question formulation to query evaluation, supporting the creation of reference query sets for any target KG.