A search for the Z gamma decay of the Higgs boson, with Z boson decays into pairs of electrons or muons is presented. The analysis uses proton-proton collision data at root s = 13 TeV corresponding to an integrated luminosity of 139 fb(-1) recorded by the ATLAS detector at the Large Hadron Collider. The observed data are consistent with the expected background with a p-value of 1.3%. An upper limit at 95% confidence level on the production cross-section times the branching ratio for pp -> H -> Z gamma is set at 3.6 times the Standard Model prediction while 2.6 times is expected in the presence of the Standard Model Higgs boson. The best-fit value for the signal yield normalised to the Standard Model prediction is 2.0(-0.9)(+1.0) where the statistical component of the uncertainty is dominant.