  We study scaling behavior of the geometric tensor
$\chi_{\alpha,\beta}(\lambda_1,\lambda_2)$ and the fidelity susceptibility
$(\chi_{\rm F})$ in the vicinity of a quantum multicritical point (MCP) using
the example of a transverse XY model. We show that the behavior of the
geometric tensor (and thus of $\chi_{\rm F}$) is drastically different from
that seen near a critical point. In particular, we find that is highly
non-monotonic function of $\lambda$ along the generic direction
$\lambda_1\sim\lambda_2 = \lambda$ when the system size $L$ is bounded between
the shorter and longer correlation lengths characterizing the MCP:
$1/|\lambda|^{\nu_1}\ll L\ll 1/|\lambda|^{\nu_2}$, where $\nu_1<\nu_2$ are the
two correlation length exponents characterizing the system. We find that the
scaling of the maxima of the components of $\chi_{\alpha\beta}$ is associated
with emergence of quasi-critical points at $\lambda\sim 1/L^{1/\nu_1}$, related
to the proximity to the critical line of finite momentum anisotropic
transition.
  This scaling is different from that in the thermodynamic limit $L\gg
1/|\lambda|^{\nu_2}$, which is determined by the conventional critical
exponents.
  We use our results to calculate the defect density following a rapid quench
starting from the MCP and show that it exerts a step-like behavior for small
quench amplitudes. Study of heat density and diagonal entropy density also show
signatures of quasi-critical points.
