The optimal or near optimal quaternary self-orthogonal codes of dimensions 2 and 3 are investigated in this paper, and the relations between the optimal self - orthogonal code length, dimension and the minimum distance are found. By means of constructing the generator matrices of such optimal or near optimal self - orthogonal codes, the optimal codes of dimensions 2 and 3 that achieve the Griesmer bound are determined. Especially, the weight polynomial of the optimal or near optimal quaternary self - orthogonal codes of dimensions 2 is obtained.