We study the mass spectrum of the quark sector in a special type-I-like model with gauge symmetry $SU(3{)}_{c}\ensuremath{\bigotimes}SU(3{)}_{L}\ensuremath{\bigotimes}U(1{)}_{X}$. By considering couplings with scalar triplets at large ($\ensuremath{\sim}\mathrm{TeV}$) and small ($\ensuremath{\sim}\mathrm{GeV}$) scales, we obtain specific zero-texture mass matrices for the quarks which predict three massless quarks ($u$, $d$, $s$) and three massive quarks ($c$, $b$, $t$) at the electroweak scale ($\ensuremath{\sim}\mathrm{GeV}$). Taking into account mixing couplings with three heavy quarks at large scales predicted by the model, the three massless quarks obtain masses at small order that depend on the inverse of the large scale. Thus, masses of the form ${m}_{u}\ensuremath{\lesssim}{m}_{d}<{m}_{s}\ensuremath{\sim}\mathrm{MeV}$ and ${m}_{c,b,t}\ensuremath{\sim}\mathrm{GeV}$ can be obtained naturally from the gauge structure of the model.