It seems that this is an open question, though I haven't been able to find many references pointing to work towards the answer.
I know that $n^2-n+1$ is conjectured to be prime infinitely often by the Hardy-Littlewood Conjecture. Moreover, a quick python script shows that about $10$% of the integers up to $n = 100,000$ yield primes under this quadratic, so this happens not infrequently.
This is a special case of Bunrakovsky's conjecture, which is only known for the degree 1 case - the famous Dirichlet's theorem. Weaker version (idealistic version) of the conjecture is true by the Chebotarev's density theorem. For example, in your case, we know that there are infinitely many primes $p$ which splits in $\mathbb{Q}(\zeta_{3})=\mathbb{Q}\left(\frac{-1+\sqrt{3}i}{2}\right)$, and in this case, $p=a^{2}-ab+b^{2}$ for some $a, b\in \mathbb{Z}$. Actually, this holds if and only if $p\equiv 1(mod 3)$ (or $p=3$), which has density $1/2$. However, your question is equivalent to whether there are infinitely many prime (or irreducible) elements in $\mathbb{Z}[\zeta_{3}]$ which has a form of $n+\zeta _{3}$, and this is an open question.