The answer is, "No"; Stephen Hawking never wrote any books about God. His focus was on his work in the area of theoretical physics and astrophysics.
Stephen Hawking's most famous book is "A Brief History of Time." He also wrote "The Universe in a Nutshell," and other books. "The Grand Design," the book discussed in the article, was his last book. He also co-authored some children's books with his daughter, Lucy. (Even the children's books were about science.)
Stephen Hawking did not write about God because his atheism was just his personal belief, based on his knowledge of science. He did not feel that it was his mission to convince anyone else about the existence or non-existence of God.
Richard Dawkins, is an evolutionary biologist. His knowledge of science also led him to a belief that God did not exist, and it could be said that his mission in life includes atheism. He wrote "The God Delusion." I can recommend that book to you if you want to understand why so many scientists are atheists.