I use Things every day and as far as I know, it doesn't have a dedicated birthdays/anniversaries feature. What I suppose you could do is make a project that's just for birthdays, and enter in separate birthdays as to-do's. If you wanted, you could then have each birthday to-do show up a few days early in the Today section, giving you advance reminders of the dates. The only way to set an alarm, though, would be to integrate that project with iCal and use that app's alarms (I think). I really like Things, but have only used it for general to-do lists, and never to set date reminders, using iCal for that.
Just out of curiosity, how come you don't want to use iCal and Address Book? 😱