To answer your question.
Theorically you might be able to. HOWEVER.......consider these thoughts and imagine you were using the 1/4" double fold bias tape. I thought of the tape as it would be already folded at 1/4". I am also assuming you are using one of the 1/4" feet that doesn't allow you to move your needle other than perhaps one click.
You would have to take extra care in making sure the edge of the quilt was firmly and completely pushed inside the binding otherwise you might not catch the quilt inside the binding.
You would need to be able to move the needle to the right so you would catch the edge of the binding. However, I don't believe the binding would alighn itself the same width on the back as it would on the front.
It might be easier if you used 3/8" folded binding and had the wider foot made for some machines that had the side guide. Viking made such a foot but am not aware the other brands have done the same.
You have posed an interesting question and I'm just trying to visualize how it might work.
If I wanted to do this, then I would consider purchasing one of the binding feet made by the machine companies that are set up especially to do both sides at the same time.
I have two such feet. One of the early ones made for Bernina (which John Flynn swears by) and the second by Marteli (which I have not used.
All the older machines (pre 1960) had a binding foot.
Anyone else have any thoughts? I'm just thinking out loud or rather with my fingers about this question. It intriegues me.