Assuming your phone uses the stock Google Contacts and Google Calendar, when adding or editing a contact, make sure to set a birthday date in one of the fields of Contacts and then save (nothing else to do in Contacts).
In Google Calendar, go to settings and check the 'Birthdays' option. Then 'refresh' the calendar (veritcal 3-dot menu on top right). You may have to repeat the refresh a few times. You should then see the contacts birthdays appear on top of Google calendar.
The birthdays appear on the calendar, but that's is all that is done. Neither the calendar on contacts apps notifies or reminds you of upcoming birthdays. You would need third party apps or automation apps to remind in advance.
Check out this birthday notification app done with the Automagic automation app:
Automagic Forum - View topic - Birthday Notification with Text, E-mail, Call - Calendar
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