An Ontario based employer will deduct and report all taxes withheld on the T4 reporting slip for Federal and Provincial purposes. Review your pay stubs and reconcile the amount of tax withheld against the amount indicated on the T4. Quebec is a higher taxed jurisdiction then Ontario and it is common in your circumstances to have a balance owing in Quebec. If the amount indicated on your pay stubs is different from the amount reported on the T4 you can contact CRA for guidance in reporting the discrepancy and CRA will contact the employer. In the meantime, you will still need to file your tax returns prior to the filing deadline.
[Note: you can transfer a federal tax refund to a balance owing in Quebec under the "Provincial" tab > "Other Quebec Credits Profile" and then "Tax Transfer for Residents of Quebec", this can allow for a balancing of the amount of refund and balance owing.]
