If you are the author of a book and want to say in an informal setting that a the book is a best seller, can you say "I'm the author of a book that has done terrific"? if not, what could be a more commonly used sentence to express the same idea?
Your sentence is correct, but a few other options sound more natural.
Specifying "sold" puts the modifier in context:
I'm the author of a book that has sold terrifically.
But if you don't want to use "sold", you can use "done well". (In certain contexts, "done well" can be a faux-modest euphemism for "made a lot of money".)
I'm the author of a book that has done well.
Avoid the issue altogether with:
I'm the author of a successful book.
Or, be specific about what the success actually means:
I'm the author of a book that sold two million copies.
"Terrific" is an adjective. You are trying to modify the verb phrase "has done", so you should use the adverb form, "terrifically": "The book has done terrifically." You could say "Sales of the book have been terrific." (Or "Critical reviews have been terrific" or "The heat from the bonfire of all the unsold copies was terrific" or whatever it is you're trying to say.)