Bayesian inversion is widely used in engineering applications, including material characterisation, structural damage identification, and medical diagnostics such as tumour detection. These inverse problems are typically ill-posed because localised variations in material properties must be inferred from sparse and noisy observations. Furthermore, they are also computationally intensive and often requiring many evaluations of high-fidelity forward solvers, such as the finite element method. Quantum computing has recently attracted significant academic and industrial interest, partly because some near-term algorithms are feasible on NISQ hardware. In particular, parameterised quantum circuits (PQCs), combined with classical optimisation, enable a hybrid quantum-classical learning. In this work, we propose a Bayesian inversion framework for inhomogeneous material properties based on a latent surrogate forward model that exploits the low-dimensional structure of spatially varying fields, such as material conductivity and the electrical potential response. The intrinsic latent dimension of these fields is identified through principal component analysis (PCA) on a training dataset. The mappings between the response field and its latent representation are learned through an autoencoder mechanism. The material property fields are projected onto a PCA latent space, and a PQC is then trained to map the latent material parameters to the latent response. We perform stochastic variational inference using this learned surrogate, operating entirely in latent space to improve computational efficiency. The methodology is demonstrated on one- and two-dimensional elliptic inverse problems for recovering spatially varying material properties.