Gozless is the 3rd city you go to, right after you come out of the Mountain Path. To be sure that the quests are active, go to the guild in Gozless and talk to the soldier at the counter - Not sure if the the quests are active until you talk to him.
Edit: thanks to
HateMe4141 for pointing out, that Quests 19-22 only appear after your first visit to Leviathan Cave.
Gozless QuestsDuty17 - Steel Fragment CollectionGather 10 Steel Fragments -
Steel Fragments are collected by checking out the blue sparkly things on the Mountain Path or in the Leviathan Cave later on.Duty18 - Small Horn CollectionGather 10 Small Horns -
Small Horns are either accquired by defeating the red/blue devil Panchos on the Mountain Path or are collected by checking out the blue sparkly things also in the Mountain Path.Duty19 - Fire Stone CollectionGather 5 Fire Stones and 3 lots of Magic Powder -
Fire Stone is accquired by checking out the blue sparkly thing in the Leviathan Cave. When you enter the cave you will see the sparkles on the right side, these are the ones you need to farm. Once farmed, press 
to leave the cave and re-enter to make them spawn again. Rinse & Repeat.
You will have more than enough Magic Powder if you fought enemies and checked out the other blue sparkles. If you don't have enough, go to the platform where the Giant Sea-serpent boss is (see Duty27).Duty20 - Ice Stone CollectionGather 5 Ice Stones and 3 lots of Magic Powder -
Ice Stone is accquired by checking out the blue sparkly thing in the Leviathan Cave. When you enter the cave you will see the sparkles on the right side, these are the ones you need to farm. Once farmed, press to leave the cave and re-enter to make them spawn again. Rinse & Repeat.
You will have more than enough Magic Powder if you fought enemies and checked out the other blue sparkles. If you don't have enough, go to the platform where the Giant Sea-serpent boss is (see Duty27).Duty21 - Thunder Stone CollectionGather 5 Thunder Stones and 3 lots of Magic Powder -
Thunder Stone is accquired by checking out the blue sparkly thing in the Leviathan Cave. When you enter the cave you will see the sparkles on the right side, these are the ones you need to farm. Once farmed, press to leave the cave and re-enter to make them spawn again. Rinse & Repeat.
You will have more than enough Magic Powder if you fought enemies and checked out the other blue sparkles. If you don't have enough, go to the platform where the Giant Sea-serpent boss is (see Duty27).Duty22 - High Wolf SuppressionSuppress 5 of the High Wolves in Leviathan Cave -
These are the purple wolves found in the Leviathan Cave. There are more than enough to fight.Duty23 - Papancho SuppressionSuppress 10 of the Papanchos around Gozless -
Papanchos are the yellow double headed Panchos found in random encounter fights around Gozless.Duty24 - Mountain Path Red DevilSuppress the Red Devil Pancho Boss on the Gozless Path -
Enter the Gozless Path from Gozless' side, use your ring, go down the stairs onto the next screen and walk down until you're in the next area. On the next screen walk down one flight of stairs and you should see a Pancho that is not moving on the right side by some trees. Aproach him and press 
to start the fight.Duty25 - Mountain Path Blue DevilSuppress the Red Devil Pancho Boss on the Gozless Path -
Enter the Gozless Path from Gozless' side, use your ring, go down the stairs onto the next screen. On the next area go down 3 flights of stairs and walk to the right side - you should then see a Pancho that is not moving surrounded by some stones. Aproach him and press to start the fight.Duty26 - Dragon's ScalesGather 5 Dragon Scales -
Dragon Scales are accquired by defeating Draco(s) on the Gozless Path or in the Sidesia Cave.Duty27 - Giant Sea-serpentSuppress the Giant Sea-serpent monster in the Leviathan Cave -
Enter Leviathan Cave go up, put the vase on the switch and go to the right side then up. Now go to platform on the left with the 2 holes in the middle - use your ring power on the yellow gems. You will see a lizard between those holes, approach it and start a fight with .When you have completed one of the above quest, you will see a bag of money with the $ sign next to the respective duty. As soon as you collect the quest reward, the bag of will change from the $ to a star sign (blue star sign means a one-time only quest). That means that you completed the quest. You can also press
go to "Quests" and select "Quests" from the menu. You will then see the exact same stats as the soldier at the guild counter will show you.
Notes:
- Re-entering rooms or "dungeons" will respawn not only the enemies but also the blue sparkly things (the blue sparkly things will only respawn if you leave to the world map!), giving you the possibility to farm the required materials for quests.
- When you completed a quest, you will still be able to complete it again for as many times you like. You just have to go to the right city guild which gave you the completed quest(s). But remember that if you collect certain materials, the amount that you had to collect for the quest will be taken away from your inventory.