The result of a search for the pair production of the lightest supersymmetric partner of the bottom quark ($\tilde{b}_{1}$) using 139 fb$^{-1}$ of proton-proton data collected at $\sqrt{s} = 13$ TeV by the ATLAS detector is reported. In the supersymmetric scenarios considered both of the bottom-squarks decay into a $b$-quark and the second-lightest neutralino, $\tilde{b}_{1} \rightarrow b + \tilde{\chi}^{0}_{2}$. Each $\tilde{\chi}^{0}_{2}$ is assumed to subsequently decay with 100% branching ratio into a Higgs boson ($h$) like the one in the Standard Model and the lightest neutralino: $\tilde{\chi}^{0}_{2} \rightarrow h + \tilde{\chi}^{0}_{1}$. The $\tilde{\chi}^{0}_{1}$ is assumed to be the lightest supersymmetric particle (LSP) and is stable. Two signal mass configurations are targeted: the first has a constant LSP mass of 60 GeV; and the second has a constant mass difference between the $\tilde{\chi}^{0}_{2}$ and $\tilde{\chi}^{0}_{1}$ of 130 GeV. The final states considered contain no charged leptons, three or more $b$-jets, and large missing transverse momentum. No significant excess of events over the Standard Model background expectation is observed in any of the signal regions considered. Limits at the 95% confidence level are placed in the supersymmetric models considered, and bottom-squarks with mass up to 1.5 TeV are excluded.