Does the Dialogue Manager have a variable with a reference to its canvas? I will like to change its values through "play mode". I have been searching for it in the documentation, but I am not really sure if I found the correct answer.
No, it doesn't. This is because the Dialogue Manager doesn't always require a canvas. If you're using NGUI or legacy Unity GUI, they don't use Canvas. And it's also possible for the Dialogue Manager to point to a dialogue UI in a canvas that's not a child of the Dialogue Manager.
However, if your Dialogue Manager has one Instantiate Prefabs component, you can find use it to get the canvas:
