  For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$
and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension
field of degree $m$. We establish a sufficient condition for existence of a
primitive normal element $\alpha$ such that $f(\alpha)$ is a primitive element,
where $f(x)= ax^2+bx+c$, with $a,b,c\in \mathbb{F}_{q^m}$ satisfying $b^2\neq
ac$ in $\Fm$ except for at most 9 exceptional pairs $(q,m)$.
