I figure if the gloves aren't wet that they have a higher chance of wetting its skin more than your own hands... so wet the gloves 
Anyhow, do you mean an American Green treefrog, or a White's/Dumpy/Australian green treefrog? If it's an American, for it to be so calm and tame is not indicative of good behavior but probably of parasites, so get it a fecal if it is! GTFs, even when used to someone's presense, shouldn't be calm and just sit on a TV or computer or anything like that.
-----
Got hips like Cinderella / Must be having a good shame / Talking sweet about nothing / Cookie I think you're Tame 