Yes, it can be upgraded. Through a long quest chain, it can be upgraded to a maximum of i135.
http://ffxiv.gamerescape.com/w... You have a very long way to go with a lot more work. The first quest (A Relic Reborn) is nothing in comparison to the entire chain... :)
http://ffxiv.gamerescape.com/w... for the i135 weapon (it only shows Dex/Vit on there because you can customize the other stats on it during the questline.)
However, I'd strongly suggest you just keep leveling and just get better weapons from Heavensward content (which also has its own relic weapon you can upgrade to a max of i270 right now). There's not a huge point to doing this questline to upgrade the weapon unless you want it for glamour purposes.