The timing of bud flush (TBF) in the spring is one of the most important agronomic traits of tea plants ( Camellia sinensis ). In this study, we designed an open-pollination breeding program using ‘Emei Wenchun’ (EW, a clonal tea cultivar with extra-early TBF) as a female parent. A half-sib population ( n = 388) was selected for genotyping using specific-locus amplified fragment sequencing. The results enabled the identification of paternity for 294 (75.8%) of the offspring, including 11 (2.8%) from EW selfing and 217 (55.9%) assigned to a common father, ‘Chuanmu 217’ (CM). The putative EW × CM full-sib population was used to construct a linkage map. The map has 4244 markers distributed in 15 linkage groups, with an average marker distance of 0.34 cM. A high degree of collinearity between the linkage map and physical map was observed. Sprouting index, a trait closely related to TBF, was recorded for the offspring population in 2020 and 2021. The trait had moderate variation, with coefficients of variation of 18.5 and 17.6% in 2020 and 2021, respectively. Quantitative trait locus (QTL) mapping that was performed using the linkage map identified two major QTLs and three minor QTLs related to the sprouting index. These QTLs are distributed on Chr3, Chr4, Chr5, Chr9, and Chr14 of the reference genome. A total of 1960 predicted genes were found within the confidence intervals of QTLs, and 22 key candidate genes that underlie these QTLs were preliminarily screened. These results are important for breeding and understanding the genetic base of the TBF trait of tea plants.