No. As calibre search can only search books in your database. And where out there is the perfect website which says for every author what books they have available?
There are two things you "can" do. The first is to create empty book entries (wishlist items effectively) in calibre for books in advance of actually buying them. Then you can either tag these especially or find them using formats:false, so you can use those as a reminder of books you want (complete with populating the published date to know when they are available).
The other thing you can do that comes to mind is to use the Search the Internet plugin, and use that to lookup an author on a site like Fantastic Fiction. You are still going to have to manually eyeball to find books you are missing to compare though.
A couple of times people have requested a plugin which would alert you to upcoming/new books for authors in their database, which I agree would be super cool. The problem is finding a source for this info. A lot of people use FictFact, but the thing I find unacceptable about them (or at least when I last looked a while ago) is they *only* store information for books that are in a series (plus only some genres). So if your author writes standalones you are out of luck.