Quenching Acid is a limited supply item that allows you to upgrade Weapons. There are 11 to find all across Fallgrim on your first playthrough, but many more available in New Game+. Weapons can be upgraded up to +5 in New Game, and then +10 in New Game+.
Quenching Acid Locations
Fallgrim Tower: Vlas the Merchant will sell you two Quenching Acid for 2500 Tar each.
Fallgrim (Fog Overworld): Locked chest near the bear traps, en route to the Fallgrim Tower
Shrine of Ash, Monument of Ash: Found inside the iron maiden in the room with the two claymore enemies, just before the large room with the flame demons.
Shrine of Ash, Monument of Ash: Found inside an iron maiden, just before reaching the final room of the temple.
Crypt of Martyrs, Abandoned Chamber: Found in a treasure chest guarded by one of the claw beasts. From the first room with multiple exits take a left and follow the path down to this room.
Crypt of Martyrs, Enshrined Sanctum: On the path leading down to the final room of the temple enter the first crawl space you see on the left side wall. You should be on a trail patrolled by ice ghosts. Find it in a chest at the end of the path.
Seat of Infinity, Eternal Narthex: Near the torch on the right side of Eternal Narthex, walk past the crossbow knights and find this chest under the rocky overhang.
Seat of Infinity, Dim Gate: Take the giant elevator near the Sester up to the top of the tower, and then take the nearby teleporter back ot the center platform. Cross the platform and take the similarly position teleporter on the other side. Go down the ramp in the middle of your destination platform, then look below where you will see a number of platforms below you. Use them like stairs, dropping down until your reach a platform with a teleporter. Take this teleporter to reach the Quenching Acid.
Seat of Infinity, Dim Gate: On the long skybridge leading to the Shifting Archives you will encounter a splitting path, two that go up and one that goes down. Follow the downward path to its end to find the Acid.
Sester Genessa: If you have purchased all the abilities on at least one Shell and area in New Game+ you can access Sester Genessa’s Glimpse shop where she will sell you 8 Quenching Acids for 10 Glimpses each.