It Probably Should Be B
The acid quasi-elemental's extraordinary ability fumes says that
The fumes from an acid quasi-elemental's body act as an inhaled poison. Creatures within 5 ft. of an acid quasi-elemental must make succeed [sic] on a DC 10 Fortitude save or take 1 point of Constitution damage each round. This poison does not have a frequency; a creature is safe as soon as it moves more than 5 ft. away from the acid quasi-elemental.
(This is what it says.)
It's pretty clear this could've used another editorial pass. Were I the editor of the Tome of Horrors Complete, I might've good-naturedly pointed out to the author that what's written says that
A creature that is within 5 ft. of an acid quasi-elemental must make a Fortitude save (DC 10). Failure means the creature suffers 1 point of Constitution damage each round forever and ever and even after that. The creature failed a Fort save! That creature sucks fumes and dies for eternity, the loser.
(This is what that means.)
Then I'd point out that the ability seems a bit much for a CR 1 Small acid quasi-elemental and recommend that perhaps a better way to put this would be that
A creature ending its turn adjacent to an acid quasi-elemental makes a Fortitude save versus poison (DC 10). Failure means that creature suffers 1 point of Constitution damage.
(This is probably what it should say.)
Luckily, the special ability fumes provides enough information that we can almost get to the revision I propose when it says there's no frequency, and, without the ability saying this is ongoing damage, this damage is a one-time effect but occurring each time a creature is within 5 ft. of an acid quasi-elemental.
The ability is contingent upon the other creature not the acid quasi-elemental, so this ability is checked when the other creature moves or stays within 5 ft. of the quasi-elemental (the DM chooses whether to have the creature make the saving throw at the ending or the beginning of the creature's turn; I suggest the ending) not when the quasi-elemental gets within 5 ft. of another creature.
Were the Constitution damage ongoing (i.e. as per your options A and C, with the character starting to take Constitution damage), the lack of a method for ending the ongoing Constitution damage would make even a single Small acid quasi-elemental an eventual deathtrap for the vast majority of creatures, making it far more dangerous than its CR 1 would suggest.