I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution about this question.

Firstly the motivation: We know that if $\gcd(a,b)=1$, then the sequence $an+b$ generates infinite prime numbers, when $n$ varies in the set $\mathbb{N}$. I want to generalize it for irreducible polynomials over $\Bbb{F}_q$, when $q$ is a prime power, i.e, $q=p^\alpha$, where $p$ is a prime number.

Let $g_i(x)\in \Bbb{F}_q[x],$$0\leq i\leq n,$ be irreducible polynomials over the field $\Bbb{F}_q$ and also let $$P(x,y)=\sum_{i=0}^{n}{g_i(x)y^i}.$$

$1)$ Is it true that if $y$ varies in the $\Bbb{F}_q[x]$, then $P(x,y)$ generates infinite irreducible polynomials over $F_q$?

$2)$ for a fixed $q$, is it possible that we choose suitable $g_i(x),$ $g_i(x)\neq g_j(x)$ when $i\neq j,$ such that the first question be true?