Given a finitely presented group (finite number of generators and finite number of relations) is there any efficient algorithms which detects whether the group is Abelian or not? (So the problem is to check whether the commutators can be generated by the relations or not.)
No, by reduction from the undecidability of checking whether the group is trivial.
Consider such a presentation. If we know the group to be non abelian, then we know it to be non-trivial. If we know the group to be abelian, them we can calculate whether it is trivial or not by computing the Smith normal form of the presentation.