So I would normally check the FAQ for this info since I know it's in there, or should be in there, but the thread seems to be completely blank for me and it's been that way for about a week hence why I am finally asking here.

How do you turn one of your current pets into a bumbus? I know in the blue building you can adopt one but I'd rather turn a current pet into one. Is this possible?

To change a pet's species you need to complete a Bumbus Queen quest, that you can get through a random event (the one that a Bumbus calls you to the BB).