First you have to create a legit motive for Q wanting to frame you to begin with. the biggest one being why he would care enough to do this. Or why he would be worried about facing some kind of federation punishment to make him need a patsy. Granted you can never be 100% but it's like how dumb it is in "Air Bud" that they let the dog play basketball. just because it's not written in the rules that dogs can't play. At some point your defense would be so unrealistic that you wouldn't need to be 100% sure.
Jason
They could say a lot of things to make up a motive like the person embarrassed Q then Q wanted to punish them by making them look bad in the eyes of their own species.
"The Q and the Grey"
Q: I'm lonely.
JANEWAY: Lonely?
Q: Oh, I know it's hard to believe, but I've been single for billions of years.
It was fun at first, gallivanting around the galaxy, using my omnipotence to impress females of every species. The fact of the matter is, it left me empty. I want someone to love me for myself. I guess what I'm saying is, I want a relationship. I just thought if you and I had a child, it would give me that kind of stability and security that I've been missing.
The story could be that they saw Q hitting on someone and getting rejected then this guy laughed and ridiculed Q afterwards. The person Q was hitting on has no memory of being hit on by Q because he wiped her memory and erased all evidence of the encounter like they erased memory here:
"Death Wish"
Q: Your Captain Honour, I am here to argue for the majesty of life. What it means to us to be alive. A Q's life takes him to all corners of the universe. This Q's life has touched and affected many, many others, including some on your own homeworld. With your permission, I would like to call some of those people whose lives have been changed by this Q.
JANEWAY: You want to bring people here from Earth?
Q: I promise it won't impact the timeline, and
no one will remember ever having being here after I send them back.
A motive can be made up and but may not be a provable motive. Or maybe there was no motive and Q was just bored like when Junior Q started a war between the Vojeans and Wyngari