I read a book that answers one of my questions, and I would like to quote it as an answer. Is that illegal?
See this article on Fair Use, which describes the ways copyrighted materials can be freely used without infringing.
The line between fair use and copyright infringement is a fuzzy one but, in general, if the amount of copied text is small, and you cite the original author, it shouldn't be a problem.
IANAL, but generally speaking and in most countries, quoting a few paragraphs from a book constitutes fair use as long as you provide a citation.
Reproducing several pages or an entire chapter would be outside the bounds of fair use.