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can you get a license in Quebec with just your work permit without a passport from your home country?
https://saaq.gouv.qc.ca/en/drivers-licences/obtaining-licence/passenger-vehicle-class-5/

Under "Eligibility", it says that if you are not a Canadian citizen, you must have proof that you are authorized to stay in Canada for at least 6 months. So if you're expecting to get a deportation order (per the title of your thread), then the answer to your question is a hard "no".
 
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