A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented. A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given, thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes. Using the relation of simple root cyclic codes and repeat root cyclic codes, the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.