A search for the decay to a pair of new particles of either the 125 GeV Higgs boson ($h$) or a second charge parity ($CP$)-even Higgs boson ($H$) is presented. The data set corresponds to an integrated luminosity of $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collisions at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ recorded by the ATLAS experiment at the LHC in 2012. The search was done in the context of the next-to-minimal supersymmetric standard model, in which the new particles are the lightest neutral pseudoscalar Higgs bosons ($a$). One of the two $a$ bosons is required to decay to two muons while the other is required to decay to two $\ensuremath{\tau}$ leptons. No significant excess is observed above the expected backgrounds in the dimuon invariant mass range from 3.7 to 50 GeV. Upper limits are placed on the production of $h\ensuremath{\rightarrow}aa$ relative to the standard model $gg\ensuremath{\rightarrow}h$ production, assuming no coupling of the $a$ boson to quarks. The most stringent limit is placed at 3.5% for ${m}_{a}=3.75\text{ }\text{ }\mathrm{GeV}$. Upper limits are also placed on the production cross section of $H\ensuremath{\rightarrow}aa$ from 2.33 to 0.72 pb, for fixed ${m}_{a}=5\text{ }\text{ }\mathrm{GeV}$ with ${m}_{H}$ ranging from 100 to 500 GeV.