  Higher-order topological insulators have been introduced in the precursory
Benalcazar-Bernevig-Hughes quadrupole model, but no electronic compound has
been proposed to be a quadrupole topological insulator (QTI) yet. In this work,
we predict that Ta$_2M_3$Te$_5$ ($M=$ Pd, Ni) monolayers can be 2D QTIs with
second-order topology due to the double-band inversion. A
time-reversal-invariant system with two mirror reflections (M$_x$ and M$_y$)
can be classified by Stiefel-Whitney numbers ($w_1, w_2$) due to the combined
symmetry $TC_{2z}$. Using the Wilson loop method, we compute $w_1=0$ and
$w_2=1$ for Ta$_2$Ni$_3$Te$_5$, indicating a QTI with $q^{xy}=e/2$. Thus,
gapped edge states and localized corner states are obtained. By analyzing
atomic band representations, we demonstrate that its unconventional nature with
an essential band representation at an empty site, i.e., $A_g@4e$, is due to
the remarkable double-band inversion on Y-$\Gamma$. Then, we construct an
eight-band quadrupole model with $M_x$ and $M_y$ successfully for electronic
materials. These transition-metal compounds of $A_2M_{1,3}X_5$ ($A$ = Ta, Nb;
$M$ = Pd, Ni; $X$ = Se, Te) family provide a good platform for realizing the
QTI and exploring the interplay between topology and interactions.
