Collision detection in 2d Quadtree

I have a quadtree with points.

Can't do two things:

1. I want the dots not to go through each other. For example, if two dots cross each other, they must as if push back.
2. I have a small rectangle and I want the dots not to go through its walls.

How do you do it in the quickest way?

I work with the quad tree like this (little pseudocode), every time I re-create it and draw (and insert dots)

function quad_init() {
   for (p in Points) {
      // move the dots a little bit to create the illusion of movement
      p.randomv_move();    
      // Collision detection here????
      tree.insert(p);
   }
}


function game_loop() {
   each tick  call function quad_init();
}