I know that horses captured by the BLM are always branded on the left side of their neck. The only breed I've ever seen that was commonly branded on the right was Standardbreds. Of course, there may be other breeds branded that way, I don't know.
If he truly is a Quarter Horse, the only thing I can think of is that it might be a breeder brand, where the "Q" is their farm, the "86" is his year of birth and the rest of it is a registration number or identification number of some sort.
Do you have good pictures of his whole body from either side? Some breeds have pretty universal appearance things that can sometimes indicate breed from pictures.