Kuakgan is an acquired status, as described in the later book Kings Of The North:
“There are other ways not to marry than cutting off an arm and grafting a tree onto your shoulder,” the Lady said, her expression grim.
“Did you not know that is what they do? Every Kuakgan, red blood with green, a tree with a limb once human, and the Kuakgan with an arm once tree. They thrive and die together.”
Kuakkganni is the plural of Kuakgan, but the term kuakgannir is used to refer to the followers of the Kuagkanni, who have no powers themselves. I don't think this is ever spelled out, but it seems clear from context in a few places, e.g. this line from Master Oakhallow in Divided Allegiance:
You aren't a Girdsman, which would put you under command of Marshal Cedfer, or a kuakgannir, which would put you under mine.
I don't believe we ever learn Master Oakhallow's race for certain, but it seems likely that he is human. He is introduced as a "tall, dark-faced man" (Divided Allegiance) and though "man" could be ambiguous, in the context of the series it seems to imply "human", for example (from Divided Allegiance again):
The teamsters are all dead, gnome and man alike.
The first quote above from Kings Of The North, depending on interpretation, could also be taken to imply that all Kuakkganni are human.