We report on a first search for production of the lightest neutral $CP$-even Higgs boson ($h$) in the next-to-minimal supersymmetric standard model, where $h$ decays to a pair of neutral pseudoscalar Higgs bosons ($a$), using $4.2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data recorded with the D0 detector at Fermilab. The $a$ bosons are required to either both decay to ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ or one to ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and the other to ${\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}}$. No significant signal is observed, and we set limits on its production as functions of ${M}_{a}$ and ${M}_{h}$.