Over the past decade, there has been a remarkable improvement in our understanding of the role of genetic variation in complex human diseases, especially via genome-wide association studies. However, the underlying molecular mechanisms are still poorly characterized, impending the development of therapeutic interventions. Identifying genetic variants that influence the expression level of a gene, i.e. expression quantitative trait loci (eQTLs), can help us understand how genetic variants influence traits at the molecular level. While most eQTL studies focus on identifying mean effects on gene expression using linear regression, evidence suggests that genetic variation can impact the entire distribution of the expression level. Indeed, several studies have already investigated higher order associations with a special focus on detecting heteroskedasticity. In this paper, we develop a Quantile Rank-score Based Test (QRBT) to identify eQTLs that are associated with the conditional quantile functions of gene expression. We have applied the proposed QRBT to the Genotype-Tissue Expression project, an international tissue bank for studying the relationship between genetic variation and gene expression in human tissues, and found that the proposed QRBT complements the existing methods, and identifies new eQTLs with heterogeneous effects across different quantile levels. Notably, we show that the eQTLs identified by QRBT but missed by linear regression are more likely to be tissue specific, and also associated with greater enrichment in genome-wide significant SNPs from the GWAS catalog. An R package implementing QRBT is available on our website.