ShareGate Desktop is designed to migrate from and to SharePoint and it has not been tested specifically for Project Management Server. However, Project Management Server has a lot in common with SharePoint. For this reason, a lot of the content from Project Management Server can be migrated with ShareGate Desktop, although the features specific to Project Server will not be migrated by ShareGate Desktop.
The Copy SharePoint Content feature should allow you to copy the content of your lists and libraries. We strongly recommend using the free 15-day trial available for download here to identify more specifically which elements of your Project Management Server can be migrated with ShareGate Desktop.