Not too long ago, at CERN, protons with relative a huge energy were smashed into one another to find out about a "large" dimension in string theory. It was conjectured that if gravity could leak into this extra dimension, a mini black hole could be created with attainable energies. The two protons, upon impact, could turn into one. No black holes were detected.
Now, assume the protons have enough energy to form a black hole. What if the protons just miss each other and fly past one another close by? Can they momentarily form a black hole in which they get stuck subsequently? A short while a high amount of energy is concentrated in a small volume.
The same holds though for a single proton though, and in that case no hole will come to be. You can always imagine a rest frame for the proton, but not for the both at once.
A moving proton can cause frame dragging. Two oppositely moving protons flying past one another closely, will give opposite frame dragging. But will their combined energy cause a black hole?
As a side note, the combined kinetic energy depends on the frame of reference. In the COM, both protons have a smaller kinetic energy as seen from one of the protons. The kinetic energy seen from the (laboratory) COM is half the kinetic energy as seen from one of the protons.
The question of which this is a supposed duplicate doesn’t refer to quantum particles, which the proton is. In too of that, Philip Gibbs, in this article, says it's not the case that a black hole forms. (Thanks to @safesphere!)