If $a,b\in\mathbb{F}_q$ with $ab\neq 0$, then the function $f=\langle a,b\rangle :\mathbb{F}^2_q\to \mathbb{F}_q$ given by $$f(x,y)=ax^2+by^2$$ is a universal quadratic form.
In order to prove the previous statement, is it sufficient to prove that every element of $\mathbb{F}_q$ is the sum of two squares?