Which is better for a kitchen with young children, quartz or granite?

Quartz, in almost every case. It is non-porous, so spilled juice, milk and sauce cannot soak in, and it never needs sealing, which is one fewer piece of household maintenance nobody remembers. It is also slightly more forgiving of impact than most granite. The trade is heat: keep a trivet by the cooker. If the household also cooks heavily with cast iron, granite becomes the better compromise and the annual seal is the price of it.