  The $\eta^{(\prime)}$-mesons in the quark-flavor basis are mixtures of two
mesonic states $|\eta_{q}\rangle=|\bar u u+\bar d d\rangle/\sqrt 2$ and
$|\eta_{s}\rangle=|\bar s s\rangle$. In the previous work, we have made a
detailed study on the $\eta_{s}$ leading-twist distribution amplitude. As a
sequential work, in the present paper, we fix the $\eta_q$ 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 $\eta_q$ leading-twist
distribution amplitude $\phi_{2;\eta_q}$ at an initial scale $\mu_0\sim 1$ GeV
are then fixed by using those moments. The sum rules for the $0_{\rm th}$-order
moment can also be used to fix the magnitude of $\eta_q$ decay constant, which
gives $f_{\eta_q}=0.141\pm0.005$ GeV. As an application of the present derived
$\phi_{2;\eta_q}$, we calculate the transition form factors $B(D)^+
\to\eta^{(\prime)}$ 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
twist-2 part, and then fix the related CKM matrix element and the decay width
for the semi-leptonic decays $B(D)^+ \to\eta^{(\prime)}\ell^+ \nu_\ell$.
