Yes they can get the Christian marriage registered under the Special Marriage Act, 1954.
Section 15 & 16 of the Act provides the complete procedure for this, please read the same:
Yes this is what provided in the Special Marriage Act,1954.
The details are also available at website of state govt e.g;
http://delhi.gov.in/wps/wcm/connect/DoIT/delhi+govt/community/marriage+certificate+and+registration
You can check yourself for your state or by visiting in person..
Or you may approach your own very able senior LOCAL counsels of unshakable repute and integrity specializing in Family/Civil matters and well versed with latest citations, LOCAL applicable rules/laws/ … and having successful track record…. and worth his/her salt…..