Ulster were the first Irish side to claim the trophy in 1999 but are now lagging behind both Leinster and Munster who have each won the trophy two times since.
Ulster also lost twice to Leinster this season and finished behind both sides in the Celtic League table but will have the chance to reverse that trend at Twickenham on Saturday.
And Trimble, part of Ireland's RBS 6 Nations side, insists he's eager to dethrone the defending champions.
"We were getting fed up of being seen as the third-choice [Irish] province," said Ireland winger Trimble.
"Playing Leinster adds intensity and passion and we want to come out on top and we also want to deny Leinster that trophy.
"Going into this final we are motivated by jealousy. We want to leapfrog them and become the number one province.
"This game means a massive amount for Ulster rugby and for Irish rugby, there is such a rivalry between the provinces.
"We have become genuine contenders for the trophy and Saturday is an opportunity we don't want to let slip."