Short answer no. Long answer nooooooooooo. Sorry, I can never resist that gag.
But let's start with the important stuff. Post the actual text.
Quenching wrote:
This special ability can only be placed on melee weapons. A quenching weapon thrust into a nonmagical fire of Medium size or smaller extinguishes it. When used against a creature of the fire subtype, it deals an extra 1d6 points of damage. The wielder of a quenching weapon receives a +2 competence bonus on saving throws against fire-based effects, and the weapon itself is immune to fire damage.
Amulet of Mighty Fists wrote:
This amulet grants an enhancement bonus of +1 to +5 on attack and damage rolls with unarmed attacks and natural weapons.
Alternatively, this amulet can grant melee weapon special abilities, so long as they can be applied to unarmed attacks. See Table: Melee Weapon Special Abilities for a list of abilities. Special abilities count as additional bonuses for determining the market value of the item, but do not modify attack or damage bonuses. An amulet of mighty fists cannot have a modified bonus (enhancement bonus plus special ability bonus equivalents) higher than +5. An amulet of mighty fists does not need to have a +1 enhancement bonus to grant a melee weapon special ability.
So there's two issues to deal with.
First, does quenching make sense to be applied to an unarmed strike (one of the requirements of amulet of mighty fists)? If not, that ends it right there. This has to be included because of rocket punches (putting throwing/returning on an amulet of mighty fists). This is, unfortunately, a GM grey area.
Second, does quenching affect the whole body? I can say with some confidence that no, no it does not, because of the ruling on amulet of mighty fists and grapple. Here's the post by the design team. The short answer is that it doesn't affect your body, it affects your unarmed strike. Which is not the same thing. You can absolutely punch a fire and put it out (and not take fire damage from that, presumably) but the rest of you would still be just as vulnerable to fire because it's not being used for an unarmed strike. It gets into a little weirder territory with natural weapons (because slam is poorly defined) but the long answer is that unless it is an "unarmed strike" or "natural weapon", it's not immune to fire. Because your arm is not an unarmed strike until it's being used to punch someone, it's not immune to fire until it's being used as a weapon. Presumably before that it's just an arm. And before you ask, yes, I do suppose you could walk around kicking the ground to make that one leg immune to fire, but that's just a little silly.
