By definition, there should be no direct involvement of the internal storage when copying data between two external storage devices, for exactly the same reasons that there should be no involvement for one external storage device when data is being copied between a different external storage device and the internal storage device. Namely, the only devices that matter are the two between which data is being copied.
However, there is at least one way it could be indirectly involved. If, for some reason, you were actively using a significant majority of the system's memory, and macOS's memory compression was unable to create enough space in RAM, you may have had to have data paged out to disk from RAM to accommodate the in-memory buffers used to copy the data from one disk to another. As the space used for paging data out to disk is almost invariably on internal storage, you would ahve thus seen a spike in disk usage when this happened.