  We obtain the well-known discrete modified Boussinesq equation in
two-component form as well as its Lax pair in $3\times3$ matrix form through a
3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We
describe how Darboux transformations and binary Darboux transformations can be
constructed for this two-component discrete integrable equation. These
transformations are then used to obtain classes of explicit solutions in the
form of Casorati- and Gramm-type determinants. $N$-soliton solutions of the
discrete modified Boussinesq equation are discussed as well when taking the
vacuum potentials as constants.
