No. You do not require or need an Internet connection (or any type of network connection) to validate any serial number. Serial numbers can be entered offline and a network connection is not required to validate the serial number. All 2BrightSparks software can be used offline.
However, for future version releases (starting from V9 onwards), it may be required to connect to the Internet to verify if Upgrade Assurance was previously purchased. If you did not purchase Upgrade Assurance, then you will not be affected by this. If Upgrade Assurance was purchased, it would be necessary for SyncBackSE/Pro to connect to our database for subscription checking and validation, but this will only happen when future versions are released.