Silicon-based storage technologies are increasingly failing to meet the explosively growing data storage demands of the information age. DNA-based data storage offers a promising solution due to its unparalleled storage density, long lifespan, low energy consumption, and high parallel accessibility. In this study, we propose a novel True Quadratic Codec System (ETQ) that directly encodes data into nucleotide sequences using a quaternary encoding approach. By treating A, T, C, and G as direct encoding symbols (0, 1, 2, 3), an ETQ eliminates the intermediate binary-to-ATCG conversion step, thus surpassing the theoretical storage density limit of 2 bits/nt. An ETQ is built on these three key components: (1) dividing image data into B, G, and R color channels for separate encoding and storage, (2) employing quaternary Huffman encoding to map image information directly into nucleotide sequences, and (3) integrating Reed-Solomon error correction codes to enhance data reliability and system extensibility. The ETQ framework demonstrates significant improvements in storage density and efficiency compared to conventional methods. By leveraging the inherent properties of DNA, this system offers a scalable and cost-effective solution that addresses the growing global data storage crisis.