For suppressing multiple access interference (MAI) in a CDMA communication system, complementary sequence sets are employed as spreading sequences in such system. In this paper, we present a method for constructing a family of quaternary periodic complementary sequence sets, which arises from the conversion of the existing binary periodic complementary sequence sets with odd period of sub-sequences. The period of sub-sequences in the proposed sequence sets is twice as long as the one of the binary sequence sets employed, which is a drawback in the proposed method. Finally, some examples are given in order to illuminate the validity of the new method.