  Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is an odd
prime power. A quadratic subspace $(W,Q)$ of
$(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ is called
dot$_{k}$-subspace if $Q$ is isometrically isomorphic to
$x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}$. In this paper, we obtain bounds for the
number of incidences $I(\mathcal{K},\mathcal{H})$ between a collection
$\mathcal{K}$ of dot$_{k}$-subspaces and a collection $\mathcal{H}$ of
dot$_{h}$-subspaces when $h \geq 4k-4$, which is given by \[\left |
I(\mathcal{K},\mathcal{H})-\frac{|\mathcal{K}||\mathcal{H}|}{q^{k(n-h)}}\right
| \lesssim
q^{\frac{k(2h-n-2k+4)+h(n-h-1)-2}{2}}\sqrt{|\mathcal{K}||\mathcal{H}|}. \] In
particular, we improve the error term obtained by Phuong, Thang and Vinh (2019)
for general collections of affine subspaces in the presence of our additional
conditions.
