Can I still become a Graphic Designer if I have done Bachelors in Fine Arts from Indus Valley School of Art?

Agree with Habib. Most graphic / web designers I know do not have any formal education related to this field. Some do courses, others learn through trail & error and online tutorials. I myself was very interested in graphics since I was young. I learned Photoshop and 3D Studio Max on my own (though it took a couple of years to get any good at it). As for developing the 'eye', without a formal education, I would recommend browsing good, aesthetic designs and especially good photography! DeviantArt is a great place for that. Oh, and joining an art community online really helps.

In no way do I mean to say this is always better than getting a formal education related to arts. However, you don't need to restrict yourself if you don't have a very specialized degree. Learn online, practice, volunteer for projects and you'll find yourself better than graduates in this field. Also, as far as I know, companies that hire designers, don't ask about their degrees IF their talent speaks for itself

Yes! Why not?
It depends upon your interest, not degree levels. You can join online classes immediately.

Creativity is an art and its in the blood of every human being. Some of them realize soon and some later. I myself, holding 10 years of designing experience, diverted from coding. And I get to this stage by only self learning no matter my degree was in computer sciences.

Thanks to lynda.com, adobe design tutorials and w3schools; which helps alot.