  Measuring bipartite fluctuations of a conserved charge, such as the particle
number, within a finite region is a powerful approach to characterizing quantum
systems. When the measured region has sharp corners, the bipartite fluctuation
receives an additional contribution known to exhibit universal angle-dependence
in 2D isotropic and uniform systems. Here we establish that the corner charge
fluctuation reveals universal information even for generic $\textit{lattice}$
systems of non-interacting electrons. We first prove that universal
angle-dependence can be recovered in the $\textit{small-angle}$ limit for
proper partitions of the lattice, from which the integrated Fubini-Study
quantum metric can be extracted. A model of a compact obstructed atomic
insulator is introduced to illustrate this effect analytically. Numerical
verification is presented for various Chern insulator models, demonstrating the
experimental relevance of the corner charge fluctuation in a finite-size
quantum simulator as a probe of quantum geometry. Last but not least, we
highlight a remarkable connection between quantum geometry and quantum
information through the lens of corner entanglement entropies.
