Isogeometric analysis (IGA) has gained significant attention for its ability to unify geometric representation and numerical approximation using spline-based basis functions. However, achieving high numerical accuracy and computational efficiency in IGA is highly dependent on the quality of parameterization, particularly when dealing with complex surfaces. We present a new approach to surface reconstruction from point cloud data that combines quasi-conformal mapping with deep learning to enable robust and adaptable surface parameterization. Although quasi-conformal theory is well-developed in the discrete setting, its use in learning-based point cloud parameterization remains limited. To bridge this gap, we propose a data-driven approach that infers quasi-conformal parameterizations from point clouds. Hence, we devise a custom loss function that incorporates suitable geometric constraints. Finally, building on this mapping, we employ advanced spline surface fitting schemes to achieve accurate and efficient surface reconstruction. This approach moves beyond the traditional paradigm of conformal mapping, highlighting the potential of integrating quasi-conformal theory with deep learning.