In this paper, we address a two-stage hybrid flowshop scheduling problem with identical parallel machines in each stage. The problem assumes that the queue (Q)-time for each job, which represents the waiting time to be processed in the current stage, must be limited to a predetermined threshold due to quality concerns for the final product. This problem is motivated by one that occurs in the real field, especially in the diffusion workstation of a semiconductor fabrication. Our objective is to minimize the makespan of the jobs while considering product quality. To achieve this goal, we formulated mathematical programming, developed two dominance properties for this problem, and proposed three heuristics with the suggested dominance properties to solve the considered problem. We conducted simulation experiments to evaluate the performance of the proposed approaches using randomly generated problem instances that are created to closely resemble real production scenarios, and the results demonstrate their superiority over existing methods. Furthermore, we applied the proposed methods in a real-world setting within the semiconductor fabrication industry, where they have exhibited better performance compared to the dispatching rules commonly used in practical applications. These findings validate the effectiveness and applicability of our proposed methodologies in real-world scenarios.