In June you will have the Southern Soul Music Festival, the Budva Music Festival and the Budva International Dance Festival. Unfortunatly I am on my phone right now and it is acting up, so you would have to google the dates. I think they all happen in the second part of June
There is also free theater plays in the old town of Budva(not sure what language as I never make it past the bars,lol) and at one square the have free concerts with local musicEdited: 4:18 am, June 10, 2018
Kotor has a open air "under the water" Film Festival in June. I can not really write a lot about it as I have never been, but it sounds interesting.
The Kotor Art Festival, that is mostly outdoors thruout the old town, which I have actually been and is rather impressive, but it starts on July 1st, so not June, but I thought to mention it if it is still in your timeframe ;-)Edited: 4:29 am, June 10, 2018
Just go to the Tourist office in Budva, they will have the most up to date info. I know the International Dance Festival/Theater are in the old town and later this month. There websites are bad, but I am sure the office will be able to give you all the info you need
Enjoy your time, Budva is amazing!