Does Matt Gaetz really have an adopted son?

No.  A year or two ago, Matt Gaetz introduced Nestor, a young man (late teens) who he described as his "son".  Nestor is not Gaetz' son, biologically or legally.  Gaetz, who is in his thirties, is not Nestor's biological father and never adopted him.  Gaetz had dated Nestor's older sister for a while.  So it's possible that they developed some sort of familial bond.  But it's my understanding that Nestor lived primarily in Cuba before moving to the US a few years ago.  At the time, a lot of people speculated that Gaetz was involved in a sugar daddy type relationship with Nestor, supporting him financially in exchange for sex.  But that's purely internet nonsense at this point.  Other people thought he was trotting out Nestor, who is Hispanic, to deflect from charges of anti-Hispanic racism against Republicans in general, and Trump in particular.  Gaetz, who is one of the most vocal Trump supporters in Congress, was seemingly trying to say that Trump can't be racist because a guy like him, with a Hispanic "son", supported him.

He is not qualified to adopt anyone.

He has no children, adopted or otherwise.