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?
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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. 

