Improving fruit quality is one of the main tasks in modern commercial apricot breeding. Because of the lack of high-density linkage maps and fine mapping, it is difficult to obtain molecular markers that can assist in breeding for quantitative inheritance of fruit quality traits. In this study, specific-locus amplified fragment sequencing was used to genotype 169 seedlings of F1 apricot (Prunus armeniaca L.) progenies derived from crossing “Chuanzhihong” (H) with “Saimaiti” (S). After aligning to the Prunus armeniaca reference genome and filtering out low-quality variants, 6,012 high-quality single nucleotide polymorphisms were obtained and employed to construct a genetic map for each parent. The genetic linkage maps showed eight linkage groups of apricot, covering a distance of 809.6 cM in “H” and 1076.4 cM in “S”. The average distance between markers in “H” and “S” was 0.62 and 0.95 cM, respectively. To map quantitative trait loci (QTLs) for fruit quality, we investigated fruit quality traits, including fruit weight (FW), fruit height (FH), fruit lateral width (FL), fruit ventral width (FV), soluble solids content (SSC), and fruit firmness (FF) for all seedlings genotyped in 2018 and 2019. Eleven and nine QTLs linked to fruit quality traits were anchored on the “H” and “S” maps, respectively, and 1,138 putative candidate genes for 16 most significant regions on the corresponding chromosome were identified based on gene annotation. Among them, fruit size contained 648 genes in 11 intervals on the reference genome, SSC contained 372 genes in 3 intervals, and FF contained 117 genes in 2 intervals. Our findings uncovered the genetic basis of apricot fruit quality, and provided candidate genes for further molecular genetic studies on fruit quality and QTL targets for future marker-assisted selection of apricot quality improvement breeding.