The Legend of Zelda: Link's Awakening
June 06, 1993
Kirby makes an appearance as an enemy in Eagle's Tower, the seventh dungeon in the game. In the international versions of the game he was renamed Anti-Kirby, due to its evil nature, leading many to believe it to be an enemy only based on Kirby, rather than actually him. Note that even if he behaves differently than the original Kirby, and is met twice in the dungeon, since his original name is "Kirby" is still an actual, official incarnation of him.
Assuming the Anti-Kirby naming, the American Nintendo Player's Guide for the game describes him as "Kirby's evil twin", in which case he would be a different character, but it would still create an in-universe link, since it would imply Kirby to exist in the Zelda universe, even though he doesn't appear in the game.
Anti-Kirby attacks by inhaling Link, just like Kirby does to his enemies, and then spit him out causing damage. It is very resistant to attacks, being only vulnerable to the Boomerang and Magic Rod (though the Magic Rod is found in a later dungeon). He can also be defeated by making him inhale a bomb.
Additionally, the Sea Urchin enemy in the game (image on the right) resembles the Gordo from the Kirby series (image on the left), but it may be a coincidence.
Kirby Super Star
March 21, 1996
One of the 60 collectable treasures in the minigame "The Great Cave Offensive", is the Triforce. It's the last treasure to be found and is also the most valuable, being worth 800000G. This is considered an actual incarnation of the Triforce since its name is the same and it's the most valuable treasure, however it doesn't seem to have the special powers that it has in the Zelda series (granting one's wishes).
Additionally, this game introduces a different hat for each of Kirby's copy abilities. The hat for the Sword ability is a green pointy hat, most likely based on Link's, except that it also has a pom-pom at the point. The sword was also slightly redesigned and the handle was recolored blue possibly in reference to the Master Sword, and Kirby also gained some new sword attacks based on ones from the Zelda series, including a chargeable the Down Thrust, and the Sword Beam, that is automatically shot when he has full health. Additionally the Bomb ability gives Kirby a blue version of the same hat, wich may be a reference to bombs being a recurring weapon in the Zelda series. These references also return in most later games in the Kirby series.
Kirby: Canvas Curse
March 24, 2002
One of the possible shapes the player has to draw on Level 3 of the Paint Panic minigame is sometimes the Triforce shape.
Kirby's Return to Dream Land
October 24, 2011
In this game Kirby gets a new ability when in his Sword form, and it's a Spin Slash very likely based on the one from the Zelda games, since it's executed in the same way and has the same effect.
Copyrights for both series are held by Nintendo.