Can you use material from a TV show when writing a book?

No because that would be a copyright violation.

The answer is a question of degree. You could write a book about a federation star ship exploring the universe without fear of prosecution. Plenty of people have done so. But if in that story you use the characters of Captain Kirk and Mr. Spock, then you will probably be charged.

Now if you ask for permission, and are willing to share your profits with those who hold the copy write then they will probably be happy to agree (especially if your writing is good), many people has written books on the Star Trek universe and characters.