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?
Ahh, after writing that post and looking in iCal, I was reminded of a longtime problem that the Birthdays subscription calendar from Address Book has - for some reason, events (i.e. people's birthdays) from that subscription do not offer you the option of setting an alarm. No idea why, but it's been a problem for a while - https://forums.macrumors.com/threads/209919/ I completely forgot about that, my bad! I think it must be an oversight on Apple's part, I mean, why else would they offer the two features and have them not work together?
If you really wanted to have alarms go off on your computer for people's birthdays, the only way I could think of is bypassing Address Book entirely, creating a project in Things with birthdays as individual to-do's, importing that project into a calendar in iCal, and setting the alarms from there.