"Is this required for the beer to be fermented properly?"
No, it should be fermented fully within a couple weeks at most, though Belgian yeast can be a bit finicky.
However the bigger issue is conditioning time. Belgian quads are usually ~11+%ABV. If you drink that two weeks after fermentation is done, it'll probably taste a bit like gasoline (a lot of 'heat' from higher alcohols). You certainly could drink it, as in it wouldn't be fermenting any more, but the flavor on a beer that strong usually doesn't peak until 4-6 months after fermentation, or longer.
Bottle-conditioning is another issue, if you choose this method. Where a lower-alcohol beer might carbonate fully in a week and a half, a beer this strong may take 3-4 weeks or more just to carbonate. Also, if it's conditioned for a number of months, you will probably have to pitch fresh yeast for conditioning to be possible.
So, really, for something you're not wanting to wait too long to drink, I think Belgian Quad, though a great style to brew, is not what you're looking for. Why not brew a Belgian Dubble, which will give you a lot of the same flavor characteristics? That should be ready within a month or two, plus you'll have a nice healthy crop of yeast you could pitch into a Quad.