Quebec does have a huge problem with birth rate, so yes, they are. (They do take on immigrants but not at anything near the rate similar socialist economies with similar demographic factors do.)

There's a cultural aspect. IMO Quebecers are more feminist than people from other places I've lived or visited. This probably gives better political support to the program, although it is not the economic reason for its existence.

But it's not inexpensive, we do have taxes to pay for it! As a working age person, I not only contribute a significant share of my salary to taxes that fund this program and others for parents, I also have to save up funds for when I am myself a parent. Yes the program and others in this vein make life generally easier for the more at-risk share of the population, but single moms in Quebec still have it pretty hard and if you're middle class you still have quite a bit to lose by having children (especially more than one).