Millimeter wave (mmWave) communications can potentially meet the high data-rate requirements of unmanned aerial vehicle (UAV) networks. However, as the prerequisite of mmWave communications, the narrow directional beam tracking is very challenging because of the three-dimensional (3D) mobility and attitude variation of UAVs. Aiming to address the beam tracking difficulties, we propose to integrate the conformal array (CA) with the surface of each UAV, which enables the full spatial coverage and the agile beam tracking in highly dynamic UAV mmWave networks. More specifically, the key contributions of our work are three-fold. 1) A new mmWave beam tracking framework is established for the CA-enabled UAV mmWave network. 2) A specialized hierarchical codebook is constructed to drive the directional radiating element (DRE)-covered cylindrical conformal array (CCA), which contains both the angular beam pattern and the subarray pattern to fully utilize the potential of the CA. 3) A codebook-based multiuser beam tracking scheme is proposed, where the Gaussian process machine learning enabled UAV position/attitude predication is developed to improve the beam tracking efficiency in conjunction with the tracking-error aware adaptive beamwidth control. Simulation results validate the effectiveness of the proposed codebook-based beam tracking scheme in the CA-enabled UAV mmWave network, and demonstrate the advantages of CA over the conventional planner array in terms of spectrum efficiency and outage probability in the highly dynamic scenarios.