The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order q are presented by La Garcia in . Being inspired by La Garcia's the paper, here we extend the results over a finite field with q(2) elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of q(2)-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in .