The $$\eta ^{(\prime )}$$ ${\eta }^{(\prime )}$ -mesons in the quark-flavor basis are mixtures of two mesonic states $$|\eta _{q}\rangle =|{\bar{u}} u+{\bar{d}} d\rangle /\sqrt{2}$$ $|{\eta }_q⟩=|\overline{u}u+\overline{d}d⟩/\sqrt{2}$ and $$|\eta _{s}\rangle =|{\bar{s}} s\rangle .$$ $|{\eta }_s⟩=|\overline{s}s⟩.$ In previous work, we have made a detailed study on the $$\eta _{s}$$ ${\eta }_s$ leading-twist distribution amplitude by using the $$D^+_s$$ $D_s^{+}$ meson semileptonic decays. As a sequential work, in the present paper, we fix the $$\eta _q$$ ${\eta }_q$ leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of $$\eta _q$$ ${\eta }_q$ leading-twist distribution amplitude $$\phi _{2;\eta _q}$$ ${\varphi }_{2;{\eta }_q}$ at the initial scale $$\mu _0\sim 1$$ ${\mu }_0\sim 1$ GeV are fixed by using those moments. The QCD sum rules for the $$0_{\textrm{th}}$$ $0_{\text{th}}$ -order moment can also be used to fix the magnitude of $$\eta _q$$ ${\eta }_q$ decay constant, giving $$f_{\eta _q}=0.141\pm 0.005$$ $f_{{\eta }_q}=0.141\pm 0.005$ GeV. As an application of $$\phi _{2;\eta _q},$$ ${\varphi }_{2;{\eta }_q},$ we calculate the transition form factors $$B(D)^+ \rightarrow \eta ^{(\prime )}$$ $B{\left(D\right)}^{+}\to {\eta }^{(\prime )}$ by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the leading-twist part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays $$B(D)^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell .$$ $B{\left(D\right)}^{+}\to {\eta }^{(\prime )}{\ell }^{+}{\nu }_{\ell }.$