A simple description of the quadrupole relaxation dynamics of atomic ions in aqueous electrolyte solution is given. It is shown that the molecular electrostatic theories often used to interpret experimental data do not directly heed to some important characteristics of the dynamics of the relaxation process. In particular, a prominent fast initial decay of the electric field gradient time correlation function is expected due to the interplay of static field gradient cancellations and conservation of correlations in the hydrogen-bonded solvation sphere. The rapid initial decay can easily lead to an order of magnitude reduction in the correlation time. This attribute of the dynamics is not specific for a particular microscopic model of solvation but should apply to any case of ionic quadrupole relaxation in a medium with strong solvent-solvent interactions. The point is illustrated by analysis of molecular dynamics simulations of aqueous solutions, considering the NMR active solutes Li-7+, Na-23+, Mg-25(2+), Cl-35-, K-39+, Br-81, I-127-, Xe-131, and CS-133+ and the paramagnetic ion Ni2+. We analyze the electric field gradient fluctuation at the solute nucleus and in each case find clear evidence for the anticipated pronounced decrease in the overall correlation time. Generally, the simulated relaxation rates, 1/T1, are in fairly good agreement with experiment. However, some inadequacies show that further refinements, such as the explicit inclusion of many-body effects, will be needed in order to achieve a universally accurate representation of ionic quadrupolar relaxation by simulation techniques.