Phase demodulation from a single moire fringe pattern is an ill-posed inverse problem that limits the applications of moire profilometry in dynamic three-dimensional (3D) measurement. In this paper, a deep-learning-based high-precision technique is used to solve this problem arising from highly under sampled inputs. Our novel approach termed two-dimensional (2D) Hilbert transformation network uses two Res U-Net networks paired with a dichotomous network to generate the desired quadrature fringe pattern by referring to the input. This process can be viewed as 2D Hilbert transformation of a fringe pattern. By using the proposed network, the wrapped phase can be extracted easily if the sampled fringe pattern is filtered and normalized in advance. Experimental results obtained using the proposed Hilbert transformation network trained on simulated data indicate that it is a simple, albeit robust solution for phase extraction from a single fringe pattern with a phase error of less than 0.02 rad. Thus, the proposed network represents a novel approach to reliable and practical learning-based single-shot Moire profilometry.