We introduce a category \mathcal O of representations of the elliptic quantum group associated with \mathfrak{sl}_2 with well-behaved q-character theory. We derive separation of variables relations for asymptotic representations in the Grothendieck ring of this category. Baxter Q-operators are obtained as transfer matrices for asymptotic representations and obey TQ-relations as a consequence of the relations in K_0(\mathcal O).