Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $K$ be a finite Galois extension of $\mathbb{Q}$ which is unramified outside the primes dividing $Np$. Let $G$ be the Galois group of $K$ over $\mathbb{Q}$. Then for any conjugacy class $C \subset G$, there is a prime number $q$ not dividing $Np$ such that $\text{Frob}_q = C$.

My question is the following: In the same situation as above, for any conjugate $C \subset G$ does there always exist a prime number $q$ not dividing $Np$ such that

(a) $\text{Frob}_q=C$;

(b) $q\not\equiv 1 \pmod p$?