Massive amounts of sequencing data are being generated thanks to advances in sequencing technology and a dramatic drop in the sequencing cost. Storing and sharing this large data has become a major bottleneck in the discovery and analysis of genetic variants that are used for medical inference. As such, lossless compression of this data has been proposed. Of the compressed data, more than 70% correspond to quality scores, which indicate the sequencing machine reliability when calling a particular basepair. Thus, to further improve the compression performance, lossy compression of quality scores is emerging as the natural candidate. Since the data is used for genetic variants discovery, lossy compressors for quality scores are analyzed in terms of their rate-distortion performance, as well as their effect on the variant callers. Previously proposed algorithms do not do well under all performance metrics, and are hence unsuitable for certain applications. In this work we propose a new lossy compressor that first performs a clustering step, by assuming all the quality scores sequences come from a mixture of Markov models. Then, it performs quantization of the quality scores based on the Markov models. Each quantizer targets a specific distortion to optimize for the overall rate-distortion performance. Finally, the quantized values are compressed by an entropy encoder. We demonstrate that the proposed lossy compressor outperforms the previously proposed methods under all analyzed distortion metrics. This suggests that the effect that the proposed algorithm will have on any downstream application will likely be less noticeable than that of previously proposed lossy compressors. Moreover, we analyze how the proposed lossy compressor affects Single Nucleotide Polymorphism (SNP) calling, and show that the variability introduced on the calls is considerably smaller than the variability that exists between different methodologies for SNP calling.