  We analyze the key origin of quantum valence criticality in the heavy
electron metal $\beta$-YbAlB$_4$ evidenced in the sister compound
$\alpha$-YbAl$_{0.986}$Fe$_{0.014}$B$_4$. By constructing a realistic canonical
model for $\beta$-YbAlB$_4$, we evaluate Coulomb repulsion between the 4f and
5d electrons at Yb $U_{\rm fd}\approx 6.2$ eV realizing the quantum critical
point (QCP) of the Yb-valence transition. To reveal the Yb 5d contribution to
the quantum critical state, we propose ultrasound measurement. We find that
softening of elastic constants of not only the bulk modulus but also the shear
moduli is caused by electric quadrupole fluctuations enhanced by critical 4f
and 5d charge fluctuations for low temperatures at the valence QCP. Possible
relevance of these results to $\beta$-YbAlB$_4$ and also
$\alpha$-YbAl$_{1-x}$Fe$_x$B$_4$ is discussed.
