No. Paying the fee starts vetting; it does not grant residence. Keep the house, the health coverage and the provincial residence until the immigrant case under EB-1 or EB-2 is granted and the move date is known.
The fee starts a process; residence comes at the end of it
The published sequence is fee, vetting, gift instruction, then the immigrant process under EB-1 or EB-2 with its own adjudication and visa availability. None of those steps has a guaranteed duration, and any of them can end the case. A retiree who sells her house on paying the fee may find herself without a Québec residence, without provincial health coverage and without U.S.
status at the same time. Keep the property and coverage until the immigrant visa is issued or adjustment is approved, and plan the move from that date. Spend the waiting period assembling the sale records and civil documents, checking the visa bulletin for the category and chargeability, and re-reading the official page before each payment, since terms can change between fee and gift.
Do not pay the gift to anyone other than the government recipient named in the official instruction, and do not pay it before that instruction arrives.