Yeah I totally agree and have always thought that about the line as well. My steelman of it is that he means “nothing” in a more handwavy way, but it’s still important to point out what you’re saying I think.
Oh, Quirrell is referring to what game theorists call Cheap Talk. If the thing I’m trying to convince you of is strictly in my own brain—like whether I intend to cooperate or defect in an upcoming Prisoner’s Dilemma—then any promises I make are, well, cheap talk. This is related to costly signals and strategic commitment, etc etc.
Anyway, I think that’s the missing piece there. “Nothing you can do to convince me [about your own intentions] [using only words]”.