So, basically, we have $c = m^e \bmod pq$, where $p$ and $q$ are both primes, $c$, $e$ and $p$ are known, and you want to determine $q$.
Obviously, we have a trivial lower bound $q > c/p$, since otherwise $c$ could not be a reduced residue modulo $pq$. However, if $c$, $e$ and $p$ are all we know, then that's basically all you can say about $q$. The key observation here is that the RSA encryption operation $m \mapsto m^e \bmod pq$ is a permutation on the message space $\{1, 2, \dots, pq-1\}$, so for each $c$, $e$ and $pq > c$ there exists an $m$ satisfying the equation.
(We can say a little more, since the permutation property only holds if $p \ne q$ and if $e$ is coprime to both $p-1$ and $q-1$, which may allow us to rule out a few candidate $q$ primes. But if $e$ is a moderately large prime, e.g. 65537, then the coprimality rarely fails.)
If we also know the plaintext message $m$, however, then in principle we can rule out most primes $q$ simply by computing $m^e \bmod pq$ for each candidate $q$ and checking if it equals the known ciphertext $c$. However, doing this checking naively by brute force would be as hard as breaking normal RSA (where we don't know $p$ a priori, but can calculate it from the known public modulus $n = pq$ and its candidate factor $q$) the same way.
Off the top of my head, I don't see any obvious shortcut for determining $q$, given $m$, $c$, $e$ and $p$, faster than by brute force. That said, I also don't see any obvious way to reduce this to a known (presumably) hard problem, so I cannot be sure that no shortcut exists. It's possible that someone more familiar with these kinds of number theoretical problems than me might spot some obvious way to apply some known theorem to settle this question. Or, for all I know, it might be an open problem waiting for someone to take an interest in it.