Hoi,
'There is a "Tamil calendar" like there is also a Malayalam, Nepali and Bengali calendar. What I have done is I have added the months. It would help when you check it out [1]. You can appreciate that I do not speak any of the languages from India :)
My advise is to first add your #babel info on your Wikidata profile. This makes life easier :) it shows you the labels in the language that you know.You can modify it from the one on my profile [2]
I am currently adding the sixty years that exist in the Telugu calendar. As I understand things, these years start on different dates compared with other calendars. As a consequence these sixty items are specific to the Telugu calendar. This means that for other languages these 60 items may need to be created as well.
Really soon now, we will be able to state how many days exist in a month ..For the Indian calendars it is much better when you add such information.The same is true for the English names; in the English Wikipedia the names are often written in a spelling that may help with pronunciation but are not written in this way by anyone I know. So please check this as well and modify to make the names more real. Obviously when you know a name for a year, a month in any of the other Indian languages please add these as well.
What you can do as well is add the Tamil name for this article [3] in Wikidata. When you do you can find it in the Tamil extended search.
Thanks,
GerardM