Another way of looking at whether (Q, +) is finitely generated or not could be as follows :
(1) All automorphisms of Q are of the form T(q) = kq (for all q in Q, some k in Q)
(2) Assume that Q is finitely generated ==> Q = < q1 , q2,....qr > having r generators
with the qi in Q.
Now, any non-identity permutation of the generators induces a
non-identity automorphism of Q.