Given two elements in $\mathbb{Q}^n \rtimes_\phi \mathbb{Z}^m$, is there an algorithm that decides if they are conjugate? Just to be explicit, $\phi$ is a homomorphism from $\mathbb{Z}^m \to Aut(\mathbb{Q}^n)$, through which elements of $\mathbb{Z}^m$ act on $\mathbb{Q}^n$. If the answer is unknown, can anyone point me in the direction of texts/papers that deal with decision problems in infinitely generated groups?
This is not an answer, but you will get an answer by doing this. Look at the paper by Noskov, http://link.springer.com/content/pdf/10.1007%2FBF01138933.pdf. As in that paper, reduce the problem to a commutative algebra problem, then use "Constructive" Commutative Algebra.

$\begingroup$ Here is a better link: dropbox.com/s/75vbn5lz1nxbni6/noskov.pdf $\endgroup$ – Igor Rivin Jul 10 '13 at 12:22