We study surface morphologies of a spherical polyelectrolyte brush in the presence of trivalent counterions using molecular dynamics simulations. Solvent quality and electrostatic interaction strength are varied to generate a series of structures. Through a careful analysis on snapshots of morphologies, shape factor of tethered chains, and monomer-monomer pair correlation function we find a nonmonotonic dependence of surface morphology on electrostatic strength, which represents clearly the electrostatic correlation effect mediated by the multivalent counterions. Due to the very importance of counterions, we further study the correlation effect by classifying counterions into four states, calculating the monomer-counterion pair correlation function and diffusion coefficient of counterions. Our simulation results clearly demonstrate that ordered patterns can be induced by the electrostatic correlation effect in the presence of trivalent counterions, which is absent in the system with monovalent ions. Also, our results can serve as a guide for rational materials design in nanoscale based on spherical polyelectrolyte brushes.