Primitive values of quadratic polynomials in a finite field

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field ${\mathbb{F}}_q$ such that $Q(g)$ is also a primitive root, where $Q(x)=a{x}^2+bx+c$ is a quadratic polynomial with $a, b, c\in $\mathbb{F}$_{q}$suchthat$b^{2} - 4ac \neq 0$.