This letter analyzes the handover probability in a drone cellular network where the initial positions of drone base stations (DBSs) serving a set of user equipment (UE) on the ground are modeled by a homogeneous Poisson point process (PPP). Inspired by the mobility model considered in the third generation partnership project (3GPP) studies, we assume that all the DBSs move along straight lines in random directions. We further consider two different scenarios for the DBS speeds: (i) same speed model (SSM), and (ii) different speed model (DSM). Assuming nearest-neighbor association policy for the UEs on the ground, we characterize the handover probability of this network for both mobility scenarios. For the SSM, we compute the exact handover probability by establishing equivalence with a single-tier terrestrial cellular network, in which the base stations (BSs) are static while the UEs are mobile. We then derive a lower bound for the handover probability in the DSM by characterizing the evolution of the spatial distribution of the DBSs over time.