  We study the scaling behavior of fidelity susceptibility density $(\chi_{\rm
f})$ at or close to an anisotropic quantum critical point characterized by two
different correlation length exponents $\nu_{||}$ and $\nu_{\bot}$ along
parallel and perpendicular spatial directions, respectively. Our studies show
that the response of the system due to a small change in the Hamiltonian near
an anisotropic quantum critical point is different from that seen near an
isotropic quantum critical point. In particular, for a finite system with
linear dimension $L_{||}$ ($L_{\bot}$) in the parallel (perpendicular)
directions, the maximum value of $\chi_{\rm f}$ is found to increases in a
power-law fashion with $L_{||}$ for small $L_{||}$, with an exponent depending
on both $\nu_{||}$ and $\nu_{\bot}$ and eventually crosses over to a scaling
with $L_{\bot}$ for $L_{||}^{1/\nu_{||}} \gtrsim L_{\bot}^{1/\nu_{\bot}}$. We
also propose scaling relations of heat density and defect density generated
following a quench starting from an anisotropic quantum critical point and
connect them to a generalized fidelity susceptibility. These predictions are
verified exactly both analytically and numerically taking the example of a
Hamiltonian showing a semi-Dirac band-crossing point.
