Harry’s classmates sit their end-of-term exams in June, finishing them on the day of the third task of the Triwizard Tournament (GF31).
The Triwizard Champions are exempt from exams. Up until the day of the task, while his classmates are being tested, Harry sits in the back of the room preparing for the third task by searching books for useful hexes (GF31).
In front of his mother and Bill, Ron claims to be fairly confident about his performance on his History of Magic exam, but admits that he made up names for some of the goblin rebels he was supposed to know (GF31).
The exams apparently take place over more than one day, so they must start no lates than June 23rd.
The History of Magic exam is after breakfast on the morning of the 24th (GF31).
June 24th (the day or the third task) is the last day of exams for Ron and Hermione, but term officially ends one week later (GF31).