I will add some some additional information.
First item
Ancient Statue drops in Lost Tower from
Death Knight.
After you receive the
Vault Expansion Certificate, right click on it to use it. Move on then to the next quest.
There are 2 more quests after this from the same NPC to do to successfully expand your inventory.
In the second part you will need to find the
Magic Cloth, which drops in Atlans from
Silver Valkyrie. For this quest item you will receive
Magic Backpack and also right click on it to use it.
In the last third part you will need to find the
Space Cloth, which drops in Icarus from
Alquamos. For this quest item you will receive
another Magic Backpack and also right click on it to use it.
After finishing all 3 quests, your inventory will be now expanded by 2 slots. Press V and click on Expanded Inventory. This is how it should look: