Can I move my bathroom sink's drain pipe about 6 inches to the right?
Yes... Maybe... It depends...

The pipe can defiantly be relocated, the question is how difficult will it be.  It may be as simple as opening the wall, cutting out the old plumbing, adding a few fittings, and closing the wall back up.  Or you might find there is framing in the way, or some other complications.

It's nearly impossible to accurately answer your question, without being able to see what you are dealing with.  The best anybody can do without seeing it, is say yes it can probably most likely maybe be moved. 

If you're not sure what to do, or don't feel comfortable doing the work yourself. Remember, your friendly neighborhood plumber is just a phone call away.