Does anyone know, where I can find a reference (preferably a book) which says that the general linear group $\text{GL}_{2}(\mathbb{Z})$ is generated by the set
$$\left\{\begin{bmatrix} 1&0\\0&-1\end{bmatrix}, \begin{bmatrix} 0&1\\1&0\end{bmatrix}, \begin{bmatrix} 1&1\\0&1\end{bmatrix}\right\}$$
Thank you in advance.
You could cite Magnus, Karrass and Solitar's book Combinatorial group theory. If I were you I would cite Theorem 3.2 (p131 - this says that certain Nielsen transformations generate $F_2$) and Corollary N4 (p169 - this says that these Nielsen transformations also generate $GL_2(\mathbb{Z})$ in a natural way: the way you want). However, there may be other "citation paths" to this result using their book, and you may be more comfortable with one of these. Certainly - the result follows easily from the book!