DNA is an excellent medium for efficient storage of information. Not only it offers a long-term and robust mechanism but also it is environmental friendly and has an unparalleled storage capacity, However, the basic elements in DNA are quaternary, and therefore there is a need for efficient coding of information in quaternary representation while taking into account various biochemical constraints involved. Such constraints create additional complexity on how information should be represented in quaternary code. In this paper, an efficient solution for the storage of JPEG compressed images is proposed. The focus on JPEG file format is motivated by the fact that it is a popular representation of digital pictures. The proposed approach converts an already coded image in JPEG format to a counterpart represented in quaternary representation while taking into account the intrinsic structure of the former. The superiority of the proposed approach is demonstrated by comparing its rate distortion performance to two alternative approaches, namely, a direct transcoding of the binary JPEG compressed file into a quaternary codestream without taking into account its underlying structure, and a complete JPEG decoding followed by an image encoding for DNA storage.