According to New Hampshire's Department of Safety if your license is suspended in any other state your license application would have been refused until your driving privileges have been restored in that former state. They do not specifically say this is true for Quebec but on your application for a license if you were asked about any restrictions or suspensions on your Quebec license that you turned in you could be in trouble if you omitted information.

To be certain that your NH driver's license is valid contact your local DMV. You can ask if violations in Quebec count on your license and even for a copy of your Motor Vehicle Record to see what is listed on it. It is easy to receive a car insurance quotation through us.