Quench detection capability is essential for reliable operation and protection of superconducting magnets, coils, cables, and machinery. We propose a quench detection technique based on sensing local temperature variations in the bulk of a superconducting winding by monitoring its transient acoustic response. Our approach is primarily aimed at coils and devices built with high-temperature superconductor materials where quench detection using standard voltage-based techniques may be inefficient due to the slow velocity of quench propagation. The acoustic sensing technique is non-invasive, fast, and capable of detecting temperature variations of less than 1 K in the interior of the superconductor cable stack in a 77 K cryogenic environment. We show results of finite element modeling and experiments conducted on a model superconductor stack demonstrating viability of the technique for practical quench detection, discuss sensitivity limits of the technique, and its various applications.