I believe the Quickstart edition does not support agents and options as such you are likely to need
1x Full Backup Exec for Windows Servers license
1x Exchange Agent License

However I also think that the Quickstart and Full version's installation files may be the same - just depends on what serial number you enter - as such you shouldn't have to unistall and re-install the product just change the serial numbers over and then insrtall the remote agent onto the Exchange server.

You might also want to bear in mind that the GRT functionality to allow efficient restore of data within individual mailboxes works best if disk based backups are used (even if you duplicate them to tape for offsite storage) as such your backup server wil need a reasonable amount of disk storage.