Pathfinding algorithm is completely separate from node storage.
Afaik a quad tree is just another form of node storage. Assuming the quadtree uses position as key, it's just another form of storing the world.
You give it a coordinate, and it returns a node to you, just like "node[pos]" in a grid-like structure, except the "[]" operation is a bit more complicated.
The pathfinding algorithm generates positions that you need to inspect. Based on the content of the queried position, you decide walkable/non-walkable, etc.
I don't know what TimeAStar does, but I assume it wants some extra information. If you can derive that information from the node contents (or from more queries of other positions), then yeah, you can use TimeAStar too.
Sorry, I have no time to try that code right now.