This paper presents a search for decays of the Higgs boson with a mass of 125 GeV into a pair of new pseudoscalar particles, H→aa, where one a-boson decays into a b-quark pair and the other into a muon pair. The search uses 139 fb-1 of proton-proton collision data at a center-of-mass energy of s=13 TeV recorded between 2015 and 2018 by the ATLAS experiment at the LHC. A narrow dimuon resonance is searched for in the invariant mass spectrum between 16 GeV and 62 GeV. The largest excess of events above the Standard Model backgrounds is observed at a dimuon invariant mass of 52 GeV and corresponds to a local (global) significance of 3.3σ (1.7σ). Upper limits at 95% confidence level are placed on the branching ratio of the Higgs boson to the bbμμ final state, B(H→aa→bbμμ), and are in the range 0.2-4.0×10-4, depending on the signal mass hypothesis. © 2022 CERN.