You need to traverse the quad-tree as follows:
1. start at root node.
2. intersect your ray with the root node's bounding box.
3. does it intersect it? if not, return (to the calling function) with no intersection (if you can't express "no intersection" in your algorithm, just use infinite distance)
4. if this is a leaf node:
- intersect the ray with all the triangles in the node, and return (to the calling function) the closest intersection
- find out which children nodes the ray intersects, and sort them according to closest intersection (this is the trickiest part in my experience)
- recurse to step 2 for each child node, in order of intersection, until the closest triangle found is closer than the distance to a child's bounding box (in which case any triangle inside that child will be further away than the one you've already found, so there's no need to traverse it)
- return (to the calling function) the closest intersection you found
This is a recursive process, and will find the closest triangle intersecting the ray with complexity O(log(n)) where n is the number of triangles in your quadtree. You probably want to implement this with a stack instead of using recursion (at least once everything is working) for performance's sake. The really annoying part is getting the bounding box collisions right, I highly recommend you overlay your quadtree on top of your triangles using wireframe rectangles to debug any issues.