The translate and line properties for 2-primitive elements in quadratic extensions

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements if, for every $\alpha,\theta\in\mathbb{F}_{q^n}^*$, such that $\mathbb{F}_{q^n}=\mathbb{F}_q(\theta)$, there exists some $x\in\mathbb{F}_q$, such that $\alpha(\theta+x)$ has multiplicative order $(q^n-1)/r$. Likewise, if, in the above definition, $\alpha$ is restricted to the value $1$, we say that $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the translate property. In this paper we take $r=n=2$ (so that necessarily $q$ is odd) and prove that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the translate property for 2-primitive elements unless $q \in \{5,7,11,13,31,41\}$. With some additional theoretical and computational effort, we show also that $\mathbb{F}_{q^2} /\mathbb{F}_q$ possesses the line property for 2-primitive elements unless $q \in \{3,5,7,9,11,13,31,41\}$.