No.  Any such polynomial would have the property that any of its restrictions $f(x)$ to one variable consist only of primes, but this is easily seen to be impossible, since if $p(a)$ is prime then $p(k p(a) + a)$ is divisible by $p(a)$.  (Even accounting for the coefficients in $\mathbb{Q}$ is straightforward by multiplying by the common denominator and using CRT; in fact, we can show that given an integer polynomial $q(x)$ and a positive integer $n$ there exists $x_n$ such that $q(x_n)$ is divisible by $n$ distinct primes.)

However, there do exist multivariate polynomials with the property that their **positive** integer outputs consist of the set of primes.  See <a href="http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem#Applications">the Wikipedia article</a>.