Yes, Spanning Backup for G Suite will back up the contents of "Computer" once they are backed up in Drive using Google's Backup and Sync app for Mac and PC. The data needs to be in Drive for Spanning to back it up.
To restore or export the backed up data from your Computer in Spanning, navigate to the user's account and click on Restore tab on the left. Choose Drive and then select "No Parent Folder". You should see the name of your Mac or PC in the No Parent Folder and from there you can select the entire folder or specific files/subfolders to export or restore. Please note that the restored content will be put back as a new folder in My Drive.