In this paper,we give a Gr?bner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an“inductive”method.Precisely,we do not use the traditional way of computing the skew-commutative relations,that is first compute all Hall polynomials then compute the corresponding skew-commutator relations;contrarily,we compute the“easier”skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first,then“inductive”others from these“easier”ones and this in turn gives Hall polynomials as a byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Gr?bner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Gr?bner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Gr?bner-Shirshov basis for the whole quantum group of type C3.