Technically, you can try, but the facility is not supposed to share that information and most staff will not.
An inmate's housing assignment, including who they share a cell with, falls under the same privacy protections as everything else about their life inside. The official position of virtually every correctional facility is that this information is not disclosed to outside parties. If you call the general line and ask who your person's cellmate is, you are almost certainly going to get a polite version of that is not information we share.
That said, relationships matter inside those buildings the same way they do anywhere else. If you have built any kind of rapport with a counselor, a unit secretary, or another staff member through regular respectful contact over time, that person might offer a small detail here and there as a courtesy. It is technically against the rules for them to do so, and you should not put them in an awkward position by pressing hard for it. But it happens, and it happens more often when the person asking is clearly coming from a place of genuine concern rather than suspicion or drama.
As a practical matter, most state prison cells house two to three people. Single occupancy is relatively rare and usually reserved for specific circumstances like protective custody or medical needs. So the baseline assumption is that your person has at least one cellmate.
The most direct and reliable way to know who he is living with is still to ask him. If he is not telling you, that is probably information worth thinking about on its own terms.