If my dependent turned 18 on June 10, 2021 but still lived with me until the end of September 2021, do I claim them on my return or do they file their own return

You can still claim them as a dependant if you are claiming credits or deductions related to them, such as a tuition transfer or medical expenses you paid for them.

Your dependant may need to file a return if they owe taxes or wish to apply for credits or deductions. For example, if they want to transfer tuition to you, they would need to first claim it on their own tax returns. This Canada Revenue Agency (CRA) page has more information on who has to file a return: Do you have to file a return?