I can’t decide. Help.
Don’t decide, look it up. It may be neither.
It’s just the “Department of Telecommunications”. Neither “India” nor “Indian” is part of the name.
If you’re writing about in a context where ambiguity is possible, you may need to qualify it, but the precise qualifier will depend on the nature of the ambiguity that you are trying to avoid. If you want to be clear that you are talking about a department of the Union government rather than a department of the government of one of the States or Territories, then maybe say “the Union Department of Telecommunications”. On the other hand, if the possible confusion is with a similarly-named department in another country, then maybe “Department of Telecommunication of India” or “India’s Department of Telecommunications” or “the Indian Department of Telecommunications”.
If you need to save space, “India’s”
If you don’t need to save space “of India” is okay
If you’re in the United States, "Indian"can be confusing. Maybe not so much in other nations.