There's no structural formula because it's what we call an amorphous solid. That is, it's like glass. It has no clearly defined crystal structure, and is likely to be extremely hard, but also extremely brittle.
As to why it's ferromagnetic or fluorescent, I'm unsure. It's likely they don't quite know, either. I suspect it has something to do with the curious bond structure of Q-carbon, which is a mishmash of both 3-way and 4-way bonds. In comparison, graphene is uniformly 3-way bonds, while diamond is a fairly uniform composition of 4-way bonds.
I speculate (don't quote me on this, I can't find a reliable source) that the way the electrons interact across the carbon atoms has something to do with why it illuminates (as photon emission is not an uncommon effect of electrons changing energy levels) and with why it's magnetic (as ferromagnetism explicitly refers to the quantum mechanical properties of electrons).
Either way. Please take this explanation with a grain of salt, but, if you're able to find more reliable sources, feel free to tack it on.