Some triliteral Arabic verbs are hollow, i.e. their middle root letter is a و or ي. The first and third root letters are sometimes repeated to form a quadriliteral verb. For instance, خاضَ يَخُوْضُ الماءَ means 'to wade through the water'. Now, from these root letters we have خَضْخَضَ 'to agitate/stir/move about/shake (water)'. Repeating the first and third root letters to form a quadriliteral verb seems to add intensiveness to the meaning, but I am not sure.
Sometimes what Ali Smith mentioned is true, but in most cases (in my opinion) the quadrilateral root has gained its independent meaning, so you use the quadrilateral root when you want to express the meaning in it. For example, قال and قلقل have little relation in meaning other than that one was derived from the other.
Also, the hollow roots are not the only ones that quadrilateral roots are derived from, they are also commonly derived from roots that have repeated letters such زلزل from زلّ, or by repeating the first two and ignoring the third such as فخفخ from فخر, or by repeating the first letter after the second such as طرطب from طرب, or by adding an unrelated letter such as شمخر from شمخ, and of course by Arabising a borrowed word such as ترجم وبرمج.
The point is, you can’t depend on a rule. These are new meanings added to the language so you use them as needed.