You are unable to submit claims using QC health cards, as Quebec does not participate in the Reciprocal Medical Billing agreement that all of the other provinces are a part of.
For Quebec billing, you will have to Invoice them as a private/uninsured patient and they would pay upfront for the services at your clinic.
The patient would have to take the invoice with the receipt and if applicable an operative report from the doctor if they underwent major surgery, to Quebec to apply to be reimbursed from their governing medical body (RAMQ).
Alternatively, you can bill directly to RAMQ and they may pay out at Quebec rates. Through ClinicAid, you could still enter it as a private bill, print and mail it to the Regie de l'assurance maladie (RAMQ).