A reddit post triggered my bullshit detector, and I figured asking the professionals would be smarter than trying to research it myself.
Hopefully, a more knowledgeable historian that spent a chunk their life researching the life of John Adams can provide clarity regarding this specific issue. In my opinion, it's a case of Brandolini's law.
There is a kernel of truth that has accreted many layers of falsehood to create a TIL "fact" that apparently has internet immortality.
The story has been debunked (five years ago!) on r/badhistory here. The truth being that there was a polar expedition that Adams supported that had a very indirect connection to the Hollow Earth proponent John Cleve Symmes, but Mole People seem to come from a joke in a Cracked article describing this story.
Anyway, I actually don't know anything about that link in the OP to History Daily, but there are some definite falsehoods in that article - John Quincy Adams most certainly wasn't elected in 1824 because people supported his backing an expedition - he backed the polar expedition four years later, and was elected by the House of Representatives in 1824 anyway. But the author's bio says she "left the world of academic", so who even knows what's going on at that site.