If you get an Ancient Heart, it means that you can draw an Ancient Weapon in the Ancient Treasure page. This can be accessed from the Ancient Weapon interface:
If you have an Ancient Heart, the ‘Summon’ button will become available, and clicking it will give you a random Ancient Weapon.
If you have an Ancient Heart, then you don’t need to go on a quest to get Ancient Weapon. You just have to click to draw the Ancient Weapon. It gives you a random Ancient Weapon, so you don’t choose what weapon you get.