No!

Proof by reductio ad impossibile.

Suppose that there is exist such polynomial .

Let now such that , when is prime.

Let now be , and let us look at:

is polynomial with integer coefficients.

So, , but is generates primes only, therefor for all integer .

Now, from the fact that polynomial can't have the same value more than times, we get the contradiction!

Here some interesting polynomials:

when then is prime...

Also(famous one):