Summarizing individual gene trees into species phylogenies using coalescent-based methods has become a standard approach in phylogenomics. However, gene tree estimation error (GTEE) arising from a combination of reasons can potentially impact the accuracy of phylogenomic inference. We, for the first time, introduce the problem of correcting the quartet distribution induced by a set of estimated gene trees, which involves updating the weights of the quartets to better reflect their relative importance within the gene tree distribution. We present QT-WEAVER, which learns the conflicts within the quartet distribution induced by a given set of gene trees and generates an updated quartet distribution by adjusting the weights accordingly. Experimental studies on a collection of simulated and empirical data sets suggest that QT-WEAVER can effectively account for GTEE, which results in a substantial improvement in the species tree accuracy. The full version of this paper is available at https://www.biorxiv.org/content/10.1101/2024.11.11.622962