This paper presents the Nordic–Estonian Quantum Computing e–Infrastructure Quest – NordIQuEst – an international collaboration of scientific and academic organizations from Denmark, Estonia, Finland, Norway, and Sweden, working together to develop a hybrid High–Performance and Quantum Computing (HPC+QC) infrastructure. The project leverages existing and upcoming classical high–performance computing and quantum computing systems, facilitating the development of interconnected systems. Our effort pioneers a forward–looking architecture for both hardware and software capabilities, representing an early–stage development in hybrid computing infrastructure. Here, we detail the outline of the initiative, summarizing the progress since the project outset, and describing the framework established. Moreover, we identify the crucial challenges encountered, and potential strategies employed to address them.