This article proposes a reduced complexity digital predistortion (DPD) scheme for fully-connected hybrid beamforming systems for massive multiple-input multiple-output (MIMO) channels impaired by nonlinear power amplifiers (PAs). Due to the inherent physical structure of the fully-connected hybrid beamforming architecture, we consider a two-stage vector DPD method to linearize the received signal at the user terminal by performing single input DPD operations at the virtual antenna elements and combining the predistorted virtual outputs properly. The required computational complexity is reduced by exploiting the spatial correlation of the antenna input signals such that redundant computation of the nonlinear complex DPD implementation is eliminated. Our results demonstrate that we can outperform the existing DPD solutions for fully-connected hybrid MIMO in terms of out-of-band emissions, the error-vector magnitude of the received signals, and computational complexity.