Let $\mathbb{F}_q$ be a finite field of odd characteristic and $2ν+δ\geq2$ an integer number with $δ=0,1$ or $2$. The orthogonal inner product graph $Oi\big(2ν+δ,q\big)$ over $\mathbb{F}_q$ is defined and the automorphism groups of $Oi\big(2ν+δ,q\big)$ are determined. We show that $Oi\big(2ν+δ,q\big)$ is a disconnected graph if $2ν+δ=2$; otherwise it is not. Moreover, we have two necessary and sufficient conditions for two vertices of $Oi\big(2ν+δ,q\big)$ and two edges of $Oi\big(2ν+δ,q\big)$ respectively are in the same orbit under the action of the automorphism group of $Oi\big(2ν+δ,q\big).$