Some breeds are much more likely to go broody than others; Henderson's chicken chart will give you an idea. Eggs start around 20-25 weeks but a hen is not likely to go broody for a while after that, and the first eggs are a little small for incubating, anyway. I have a couple of mutts that went broody around 8 or 9 months.
If you want to be fairly certain of getting a broody, Old English Games, Kraienkoppes, Silkies, and some other game breeds such as Asil are the most likely. Other more common breeds such as Cochins, Buff Orpingtons and Black Australorps are fairly likely to go broody. Occasionally you will get a broody in most any breed, though.