Answer
Yes, links in pages, lists, and libraries will be updated to reflect the new destination URL (though content will not be changed).
If desired, you can disable the automatic link updates when copying content. Otherwise, this operation cannot be customized.
Limitations
ShareGate Desktop cannot correct URLs within documents. For more on this limitation, see URLs that are contained in documents will not be corrected.
ShareGate Desktop does not correct URLs for terms with navigation node type Simple Link or Header. For more on this limitation, see Term navigation limitation when using Simple Link or Header.