Passport stamps are only issued on entry into the island of Ireland through an airport or port.
The border between the Republic of Ireland and Northern Ireland is unmanned as far as customs is concerned, most people only realize they have crossed the border by the speed limit signs changing from Kilometres per hour to Miles per hour!
Perhaps you could contact the UK Home Office
to see if you can apply for a stamp. If you are entering in Dublin you will receive a stamp of entry for the Republic of Ireland which is valid during your whole stay within the Common Travel Area.
Maybe some of our community members will have some more helpful information too.
Where have you planned to visit during your time in the Belfast