The short answer is yes. QIIP leads to permanent residency in Canada, hence upon your "landing" in Canada, you would be protected by Article 6 of our constitution which guarantees the right to live and work freely throughout the country. The above being said, you must have an intent to settle in the province of Quebec throughout the process of QIIP and upon activation of your status of permanent resident. For this reason, we strongly suggest to try your best to establish yourself in the province of Quebec after getting your permanent resident visas. If it does not work after some time, you would be able to leave and go to another province. In any event, if you establish yourself in Quebec, you can of course travel to other provinces at your will.
Do the Canadian immigrant investor programs allow the mobility to live throughout the country?
I am pursuing the Quebec program, but am wondering after I establish residency there, can I live in other provinces, either full time or part time? Will leaving Quebec abandon my ability to live in the country?
A successful application to the QIIP program grants the applicant (as well as spouse and children) permanent residency, which is federally (throughout Canada) recognized status. With that being said, the program exists to benefit the province of Québec and its SMEs. As such, the applicant is expected to settle in Québec, and is required to sign an agreement along with the application stating this intention. A PR status allows you free movement between the provinces, but settling in location other than Québec undermines the raison d'être of the program.
The QIIP is meant for people who have intention to go to Quebec. If you do not have a real intention, it could be considered as a misrepresentation.