I think that, in this generality, the answer is no.
For instance, take $a=4$, $b \geq 3$. Then $p \geq 7$, so $a=2^2$ is a square mod $p$.
But a generator $x$ of $(\mathbf{Z}_p)^{\times}$ is always a non-square: indeed, if $x=y^2$ then $x^{(p-1)/2}=y^{p-1}=1$, so the order of $x$ divides $\frac{p-1}{2}$, in particular it is strictly less than $p-1$.