Here's a solution using the answers package.
You can toggle the answers appearing or not by using
\usepackage{answers}
to write the answers to a file (answers.tex in the code below) or
\usepackage[nosolutionfiles]{answers}
to show the answers in the document.
I've used the enumitem package to create a list-type environment called questions, and defined the \question command as \item (although it doesn't take an optional argument- not sure if this matters or not here though).
\documentclass{amsbook}
\usepackage{enumitem} % for custom lists
\usepackage{answers} % for separating solutions
%\usepackage[nosolutionfiles]{answers}
\newlist{questions}{enumerate}{5}
\setlist[questions]{label*=Question \arabic*.}
\newcommand{\question}{\item}
% open the answer file
\Opensolutionfile{answers}
\Newassociation{answer}{ansWER}{answers}
\begin{document}
\begin{questions}
\question $1+1=?$
\begin{answer}
$1+1=2$
\end{answer}
\question What is $1+2=?$
\begin{answer}
$1+2=3$
\end{answer}
\end{questions}
% close the answer file
\Closesolutionfile{answers}
% use this line if you'd like to include the answers separately
\clearpage
\IfFileExists{answers.tex}{\input{answers.tex}}{no file!}
\end{document}
You can do a lot with the answers package, particularly if you combine it with the hyperref, as shown in a few answers on this site, for example in Adding another answer hyperlinked to the question itself