A search for a heavy neutral Higgs boson, , decaying into a boson and another heavy Higgs boson, , is performed using a data sample corresponding to an integrated luminosity of 139 fb from proton-proton collisions at TeV recorded by the ATLAS detector at the LHC. The search considers the boson decaying into electrons or muons and the boson into a pair of -quarks or bosons. The mass range considered is 230-800 GeV for the boson and 130-700 GeV for the boson. The data are in good agreement with the background predicted by the Standard Model, and therefore 95 % confidence-level upper limits for \sigma\times B(A\to ZH)\times B(H\to bb \; \textrm{or} \; H \to WW) are set. The upper limits are in the range 0.0062-0.380 pb for the channel and in the range 0.023- 8.9 pb for the channel. An interpretation of the results in the context of two-Higgs-doublet models is also given.