You can sell a weapon with 20% quality to an NPC to get 1 Blacksmith's Whetstone. You can also sell several weapons with total quality of at least 40% to get one. Those methods are likely never worth it, as Blacksmith's Whetstones are nearly worthless.
By the end game these will be so common loot drops, that you will find more than you will ever need
Blacksmith's Whetstones are very cheap, so use them as you like, although you possibly won't have enough of them to use on every item while leveling.
You can increase the quality of a weapon to up to 20%. Quality only increases the Physical Damage of a weapon - shown on the first line in the item description.
Some characters, like spellcasters, don't care about the weapon's physical damage at all, in which case quality will give no benefit.
One Blacksmith's Whetstone adds 1% quality to a Rare or Unique item, 2% quality to a Magic item, and 5% quality to a Normal rarity item. So if you are crafting something, it is most efficient to add quality before you turn the item into a higher rarity.
In the end game, Blacksmith's Whetstones can be used in creating another, more valuable currency - Cartographer's Chisels. If you increase the quality of a Stone Hammer, Rock Breaker or Gavel (types of one handed Mace) to 20% and sell it to an NPC along with any Map item, you will get 1 Chisel in return.