It really depends on how long you're planning to stay in Quebec, and how likely your older child is to pick up the language quickly.
What the others say is mostly correct. The English public schools are primarily for historical Anglophones (English speakers) who are from Canada. So if you're from anywhere in Canada, and you went to school in English, then your child is therefore eligible to go to school in English. But if you went to school in English anywhere else (US, England, Australia), then your child is not eligible, and must go to school in French (unless you want to pay for private).
As a person here on a temporary work permit, your children can attend school in English. If you are coming on the work permit for three years, then leaving, it may be worth it to go through the hoops required to get your kids to be able to go to school in English.
However, even if you do that, if you decide to become PR, then your children will have to switch to a French school, unless you want to pay for private English school. It may make sense, depending on your circumstances and the timing of it, to have your older child go to school in English (public), and then wait to file for PR until your older child has finished school. But your younger child won't finish school by the time you're applying for PR, so for them, unless you want to pay for private English school, you can enroll them in French school. (Although one thing to think about is if you hope that your family, including your older child, will really settle permanently in Quebec, it would be very difficult for them to find a job without any French.)
Feel free to DM me about any of this, I've been here for 3.5 years, and have a 6 year old child, so I've done a bit of research into this.
