@Noah S Thanks for posting your query on Microsoft Q&A.
If both VMs are in the same region and availability zone, the likelihood of a background copy is reduced, but not entirely eliminated. The disk might still be moved between different storage locations within the same region, which can trigger a background copy.
Changing the storage type (e.g., from Standard HDD to Premium SSD) will likely involve a background copy, regardless of the region or availability zone.
For managed disks, Azure handles the storage location and type automatically. If the managed disk needs to be moved to a different storage location for any reason, a background copy might be triggered.