This paper presents a search for decays of the Higgs boson with a massof 125 GeV into a pair of new pseudoscalar particles, H -> aa, where onea-boson decays into a b-quark pair and the other into a muon pair. Thesearch uses 139 fb(-1) of proton-proton collision data at acenter-of-mass energy root s = 13 TeV recorded between 2015 and 2018 bythe ATLAS experiment at the LHC. A narrow dimuon resonance is searchedfor in the invariant mass spectrum between 16 GeV and 62 GeV. Thelargest excess of events above the Standard Model backgrounds isobserved at a dimuon invariant mass of 52 GeV and corresponds to a local(global) significance of 3.3 sigma (1.7 sigma). Upper limits at 95\%confidence level are placed on the branching ratio of the Higgs boson tothe bb mu mu final state, beta(H -> aa -> bb mu mu), and are in therange 0.2-4.0 x 10(-4), depending on the signal mass hypothesis.