The proliferation of Internet of Things (IoT) devices equipped with acoustic sensors necessitates robust acoustic scene classification (ASC) capabilities, even in noisy and data-limited environments. Traditional machine learning methods often struggle to generalize effectively under such conditions. To address this, we introduce Q-ASC, a novel Quantum-Inspired Acoustic Scene Classifier that leverages the power of quantum-inspired transformers. By integrating quantum concepts like superposition and entanglement, Q-ASC achieves superior feature learning and enhanced noise resilience compared to classical models. Furthermore, we introduce a Quantum Variational Autoencoder (QVAE) based data augmentation technique to mitigate the challenge of limited labeled data in IoT deployments. Extensive evaluations on the Tampere University of Technology (TUT) Acoustic Scenes 2016 benchmark dataset demonstrate that Q-ASC achieves remarkable accuracy between 68.3% and 88.5% under challenging conditions, outperforming state-of-the-art methods by over 5% in the best case. This research paves the way for deploying intelligent acoustic sensing in IoT networks, with potential applications in smart homes, industrial monitoring, and environmental surveillance, even in adverse acoustic environments.