Just like any other marks, you can apply to register the website name as well. One thing you need to decide upon is if you want to register with .in or .com or any other extensions.
Like any other trademark application, this will go through various steps like trademark search, application and its examination, publication in trademark journal, opposition (if any) and finally the registration. You can refer to the detailed trademark registration process[1] from the link in footnote.
The first step you need to take if consult a trademark attorney to take the availability search of the mark. Based on the search report, you can decide to apply for registration or not.
Footnotes