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?



irreducible polynomials on the polynomial sequence
If $g\_1(x)$ and $g\_0(x)$ are relatively prime, then there are infinitely many irreducibles of the form $g\_1(x) y + g\_0(x)$. This is due to Kornblum and Landau (1911) and recorded in many places; see Chapter 4 of [Number Theory in Function Fields](http://books.google.com/books?id=vDpa_C5DIbkC&pg=PA193&lpg=PA193&dq=%22number%20theory%20in%20function%20fields%22&source=bl&ots=subu56C3mW&sig=OHbk2iPbaWDdOJu4Y10npM8M8DE&hl=en&sa=X&ei=1zLEU7_KFI3A7Abcu4HgDA&ved=0CD8Q6AEwBQ#v=onepage&q&f=false) for a clear modern exposition.


For $G(x,y) = \sum\_{i=0}^n g\_i(x) y^i$ for $n>1$, the obvious necessary conditions are that 
$G(x,y)$ be irreducible as a polynomial in $2$ variables, and that there is no $\pi(x)$ such that $\pi(x) | G(x, f(x))$ for all $f(x) \in k[x]$. In the integer case, these conditions are conjectured to be sufficient. In the function field case, they are not! See [Conrad and Conrad and Gross](http://www.math.uconn.edu/~kconrad/articles/genuszero.pdf). Their exposition is very clear, and they conjecture what sufficient conditions should be. 


In a positive direction, [Pollack](http://plms.oxfordjournals.org/content/97/3/545.full.pdf+html) shows that, if you fix $G(x,y)$ in $\mathbb{F}\_q[x,y]$ avoiding the obvious obstructions and with $GCD(\deg\_y G, q)=1$, fix a degree $n$, and let $y$ range over degree $n$ polynomials with coefficients in $\mathbb{F}\_{q^a}$, then the number of prime values obeys the expected asymptotics (and, in particular, is positive) as $a \to \infty$. I am simplifying his result some for exposition; see the original paper.


I don't understand the second part of your question.


