It turns out that blue pigment, in animals, is something that hasn't been figured out by evolution. A quick stroll over to Wikipedia shows a number of animals who quite obviously have blue parts. How is this done, then?
Well, in the case of dogs, "blue" colors are just a slightly special form of grey.
They're cute, but not truly blue. Just really grey. So let's move on.
There is the case of feathers, butterfly wings, and maybe some other animals, they actually don't use blue pigment at all. They use something called structural coloration. If you're a physicist, you call this a special case of wave interference. Whatever you choose to call it, it works by reflections of light amplifying (or suppressing) particular wavelengths of light. A brief schematic looks like this:
So, this means our Bluebeard can have a weird hair-color gene or very, very particular hair follicles. Either method would explain naturally blue hair without dyes.
The final option is that Bluebeard's body somehow alters his hair proteins to look blue. If this were the case, Bluebeard's blueness would depend on what he eats or drinks, so he could (theoretically) go on a diet to "deblue" himself.
In any case, a blue beard would have been quite the mutation! So crazy, in fact, that I would suspect genetic engineering. So here is your succinct answer: Yes, it's within the realm of physical possibility to have a naturally blue beard, but no, not without something extraordinary going on.