Yes, but perhaps not in the way the question implies.
I review books, and a lot of the books I review are sent to me directly by the author. In these cases, I always contact the author after reviewing the book to share my view. (Not least because showing I reviewed their book will hopefully ensure I get the opportunity to review their next book as well.)
Sometimes I don’t enjoy a book enough to recommend to my readers.
In these cases, I’ll send the author my review and ask if they want me to post it. Most say yes (which is the professional response), and some ask that I don’t post my review as soon as the book releases. I don’t have a problem with that: while reviews need to be honest and it’s okay to honestly not enjoy a book, there’s no need to advertise the fact by being the first Amazon or Goodreads review.
I have had authors contact me to thank me after I’ve posted a review for their book, and that’s always nice. But that’s the author contacting me and me responding. Not me initiating the contact.
I have signed up for more than a few author newsletters, and authors will often ask a question in their newsletters. I will sometimes reply to author newsletters, but that’s not really a result of reading their book.
Apart from that, have I ever contacted an author after reading their book?
I don’t think so, although I know authors appreciate hearing from their readers. I prefer to review or talk about a book I’ve enjoyed.
