20 cards where the text includes “quest counter” and it’s legal in Commander and the color identity ≤ BGUW

Beastmaster Ascension (New Capenna Commander #283)
Bloodchief Ascension (Commander Masters #139)
Khalni Heart Expedition (Commander Masters #899)
Luminarch Ascension (Masters 25 #23)
Quest for Renewal (Worldwake #110)
Quest for Ula's Temple (Worldwake #35)
Archmage Ascension (Zendikar #42)
Quest for the Nihil Stone (Worldwake #64)
Ior Ruin Expedition (Commander Legends #398)
Yes Man, Personal Securitron (Fallout #29)
Quest for the Holy Relic (Zendikar #33)
Overseer of Vault 76 (Fallout #19)
Quest for the Gravelord (Battlebond #156)
ED-E, Lonesome Eyebot (Fallout #131)
Soul Stair Expedition (Zendikar #112)
Arcade Gannon (Fallout #92)
Sierra, Nuka's Biggest Fan (Fallout #25)
Quest for Ancient Secrets (Zendikar #59)
Sunspring Expedition (Zendikar #37)
Quest for the Gemblades (Zendikar #177)

Though this query has ended, its relics still slumber in New Argive.