Enterprises and online merchants can benefit from automatically detecting complaints from social media platforms and shopping websites. Prior studies on automatically identifying complaints on social media have relied chiefly on analyzing complaints in English, with little consideration for code-mixed complaints. We manually annotated the emotion and sentiment classes in the Product Review dataset, a code-mixed language complaint corpus. The extended corpus consists of 3711 instances annotated with the complaint, sentiment, and emotion labels. We present a Graph Attention Network (GAT) based multi-task framework that intends to simultaneously learn three closely related tasks, complaint detection (primary task), sentiment classification, and emotion recognition (auxiliary task). Based on a quantitative and qualitative study on the extended version of the Product Review corpus, the proposed approach attains an accuracy of 72.82% and a macro-F1 of 71% for the intended task of recognizing complaints, outperforming several single-task, multi-task baselines.11Resources are available at: https://github.com/appy1608/ESWA2022_GraphIC.