June is the beginning of monsoon in many parts of India, that allows lot of fresh air, fresh greenery giving lush nature view. Here are a few to visit
Munnar - Asthamudi in Kerala
Coorg and Kudremukh in Karnataka Mahabaleshwar and Panchgani in Maharashtra Nainital and Panchgani in Uttarakhand Darjeeling and Pelling and Meghalaya in North east
Auli- Valley of flowers -Auli-Chopta -Kasol -Dalhousie-Mcleodganj are some more hill stations to explore in Himachal
Manali Shimla Gulmarg sonamarg and Pahalgam in Kashmir
I would suggest you to visit a hill station before June during April and May because there are chances of rain during June which might cause landslide. But if you want to visit a hill station I would suggest you to visit Kalimpong in West Bengal.