As Jeremy mentioned in a comment, your question is still open. I merely add that what is conjectured is that if an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ satisfies $1=\mathrm{gcd}\{f(1), f(2), f(3), f(4), \dots\}$ then $f(n)$ is prime for infinitely many $n$. This is known as Bunyakovsky's conjecture. It has not been proven for any polynomial of degree greater than $1$. Generalizations include Schinzel's hypothesis H and the Bateman-Horn Conjecture.

As far as I am aware, it is unknown whether any irreducible polynomial of degree greater than one assumes infinitely many prime values. Certainly this is the case if one insists that the polynomial be given explicitly. I merely add that what is conjectured is that if an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ satisfies $1=\mathrm{gcd}\{f(1), f(2), f(3), f(4), \dots\}$ then $f(n)$ is prime for infinitely many $n$. This is known as Bunyakovsky's conjecture. It has not been proven for any polynomial of degree greater than $1$. Generalizations include Schinzel's hypothesis H and the Bateman-Horn Conjecture.