Abelian groups are the same thing as $\mathbb Z$-modules. In general, for any ring $R$, the theory of left $R$-modules has quantifier elimination down to Boolean combinations of primitive positive formulas and certain sentences (expressing so-called Baur–Monk invariants). This is the Baur–Monk quantifier elimination theorem; see e.g. §A.1 in Hodges, Model Theory.
In particular, since the theory of any particular abelian group or $R$-module is complete, it has quantifier elimination down to Boolean combinations of p.p. formulas.