No, King Charles III does not rule over both Quebec and the UK. King Charles III is the current heir to the British throne, and he will become the king of the United Kingdom once his mother, Queen Elizabeth II, abdicates, retires, or passes away. However, Quebec is a province in Canada, and it is not part of the United Kingdom. Quebec is governed by its own provincial government, led by a Premier, and it is part of the larger country of Canada, which has its own system of governance and a separate head of state, the monarch of Canada. So, while King Charles III will be the king of the UK, he does not have any direct rule or authority over Quebec.