Hughes said the 34-year-old 'keeper had sent text and phone messages to QPR technical director Mike Rigg and chairman Tony Fernandes.
QPR made contact with Leeds to notify them about the messages on Saturday.
But Hughes said: "It doesn't go anywhere from here."
He added: "There were some text messages and telephone messages, which were to Tony Fernandes and our technical director, Mike Rigg.
"We felt they were inappropriate. We didn't make a formal complaint. I think Mike just got in touch with Leeds just to make them aware of the situation and just to acknowledge that actually it did occur.
"But we didn't make a formal complaint [to the football authorities]. It was just two individuals."
Neil Warnock has refused to comment on the situation.
He said: "We have a club policy on Twitter and social networking. It's an internal matter and we will not be commenting further."
Kenny has also got into a number of disagreements with QPR fans on Twitter over the last few days and referred to them as a "tin pot club" in one tweet.
The former Republic of Ireland international made 77 league appearances in two years at Loftus Road and helped them to promotion to the Premier League in 2011 before joining Leeds in the summer.