Semantic communication (SC) enables bandwidth-efficient coordination in multi-agent systems by transmitting meaning rather than raw bits. However, when agents employ heterogeneous sensing modalities and AI architectures, perfect bit-level transmission no longer guarantees mutual understanding. Although deep learning methods for semantic compression have advanced, the information-theoretic limits of semantic alignment under heterogeneity remain poorly understood. Notably, semantic ambiguity shares the same mathematical structure as quantum contextuality, as both arise from cohomological obstructions, motivating a quantum formulation of SC. In this paper, an information-theoretic framework for quantum semantic communication is proposed using sheaf cohomology. Multi-agent semantic networks are modeled as quantum sheaves, where agents meaning spaces are Hilbert spaces connected by quantum channels. The first sheaf cohomology group is shown to characterize irreducible semantic ambiguity, representing a fundamental obstruction to alignment that no local processing can resolve. The minimum communication rate required for semantic alignment is proven to scale with the logarithm of the dimension of the cohomological space, establishing a semantic analog of Shannon limits. For entanglement-assisted channels, the achievable capacity is shown to strictly exceed classical bounds, with each shared ebit reducing the required classical communication by one bit, providing a rigorous interpretation of shared context. Additionally, quantum contextuality is shown to reduce cohomological obstructions, and a duality between quantum discord and integrated semantic information is established, linking quantum correlations to irreducible semantic content. This framework provides rigorous foundations for quantum-enhanced semantic communication in autonomous multi-agent systems.