To unlock the Uncommon Talent trophy in Anthem, you will need to increase the Gear Score of one of your Javelin's until it reaches the uncommon level. Simply equipping uncommon items in your five slots is not enough. The overall Javelin's level must be uncommon.
The gear score is an average of all of your gear. The higher the score, the higher the rarity. Once you have reached the uncommon level with a Javelin, complete a mission, contract, or stronghold and the trophy will unlock at the end.