Means the same as, "I don't give a flying fuck", "I don't give a Funyun", "I don't give a
rat's
patootie", etc.
This is a clean
way of saying it -- though that hankie really needs to go in the wash. :-)
WarGrowlmon: What? Beelzemon?
Taomon: What are you doing here?
Rapidmon: Looking to hit us while we’re down, huh? That’s your style right?
Beelzemon: Hey,
trash me if you want but I swear I’m here to help.
Rapidmon: Do you even know the meaning of the word?
Beelzemon: I’m on the level fellas, really.
WarGrowlmon growls at him.
Beelzemon: But I don't give a dirty handkerchief if you believe me or not, cause my partners believe in me.
Flashback to
Mako giving Impmon his
gun.
Mako: You can blast all the bad guys with it.