She should be able to start suggesting the first day you see her outside. It's happened to me before, but it is very rare, especially since you probably don't have much friendship with her.
Whenever RNG favours you with its benevolence. Once you get past the introduction stage, theoretically a villager can request a PWP at any time; I don't know if it's 100% true that friendship plays a role, but based on personal, non-experimentally verified anecdotal evidence, I'd say that it does. So...time for some fetch quests!