Yes, you can go to Pangong Lake and Nubra Valley in June. You will be able to see all of the main attractions of the area if you have a week to spend in Ladakh in June. After seeing the nearby monasteries and enjoying the city of Leh, continue to Pangong Lake in the Nubra Valley. The last of the month during the Hemis festival is the best time to visit Leh Ladakh in June.