Yes you can, but be aware that there are very few breeds which you can keep as legal pets in India and they all need a lot of care and maintaince, irrespective of general opinion and belief. Also you can not keep any animal as a pet if it's natural habitat is in the Indian wilds. read more
Keeping turtles as pet in India is illegal in case of native turtle species, however u can pet a Red Eared Slider (RES) turtle. Which is one of the most common pet turtle all across the world and is also legal to pet in India, since it is not a native species. You can check for the RES turtle in the nearby Fish and Aquarium sellers. read more