Azamara Quest has no children's areas or programming, though kids are allowed onboard. Infants must be 6 months to sail (or 12 months on all transoceanic voyages). Although the ship is not set up for kids, we saw more families than we expected, including an infant and several school-age kids, as well as a few teens.
The pool, Wii, some board games and in-cabin TV's are pretty much the only shipboard amenities that will entertain children. If you plan on bringing your brood onboard, make sure they're capable of finding their own fun while on the ship. Despite the lack of children's programs, the staff did seem to go out of their way to reach out to the youngsters; I noticed this especially in the Windows Cafe, where the servers would call to the children by name and engage them in conversation.