Inmates are not permitted direct internet access and cannot personally operate a Facebook or any other social media account from inside a correctional facility. What is happening is one of two things. Either someone on the outside is managing the account and posting or messaging on the inmate's behalf, or the inmate has access to a contraband device and is operating it themselves in violation of facility rules.
Both situations are serious and both give you a path to address it.
Start by calling the facility and asking to speak with the warden's secretary. Explain that you are receiving harassing messages through Facebook Messenger from an account associated with this inmate, provide the specific details including the account name and the content of the messages, and request that the matter be investigated. Make clear that if the facility does not respond you will be taking the issue to local law enforcement. That combination typically gets attention.
If a contraband phone is involved, reporting it to the facility is exactly what they need to know about. Unauthorized devices inside correctional facilities are a serious security issue and facilities investigate these reports. The inmate faces significant disciplinary consequences if a device is found.
On your end, document everything before reporting. Screenshot every harassing message with timestamps before doing anything else. That documentation is essential whether you are reporting to the facility or filing a police report.
You can also report the Facebook account directly to Meta for harassment, which can result in the account being restricted or removed regardless of who is operating it.
Do not engage with the messages. Block the account, preserve the evidence, and report through both channels simultaneously.