  Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd
prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in
\mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in
\mathbb{F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace
x\in \mathbb{F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{F}_q^d, r\in
\mathbb{F}_q$. We prove bounds on the number of incidences between a point set
$\mathcal{P}$ and a $Q$-sphere set $\mathcal{S}$, denoted by
$I(\mathcal{P},\mathcal{S})$, as the following.
  $$| I(\mathcal{P},\mathcal{S})-\frac{|\mathcal{P}||\mathcal{S}|}{q}|\le
q^{d/2}\sqrt{|\mathcal{P}||\mathcal{S}|}.$$
  We prove this estimate by studying the spectra of directed graphs. We also
give a version of this estimate over finite rings $\mathbb{Z}_q$ where $q$ is
an odd integer. As a consequence of the above bounds, we give an estimate for
the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the
number of incidences between a random point set and a random $Q$-sphere set in
$\mathbb{F}_q^d$. We also study the finite field analogues of some
combinatorial geometry problems, namely, the number of generalized isosceles
triangles, and the existence of a large subset without repeated generalized
distances.
