Well, you could, but you'd have to carb it at higher pressure. There's a couple of things that you could do- you could prime the keg, and then leave it to condition at room temperature. The downside of that is more sediment in the keg, and when you move it to the kegerator it'll stir up a little bit. It works, though, especially if space is tight.
For fairly young beers, I use the keg as a secondary (sort of). I keg it, then purge the headspace with co2 and seal it up with a shot of co2 and keep it at room temperature for about 2 weeks. By that time, the beer is only 4-5 weeks old. Then, I move it to the kegger and set it at serving psi (I use 12 psi). It takes about a week to carb up. That means the beer is about 6 weeks old when carbed and on the tap. Still young, but for a low OG beer, it's ok to drink it.
The temptation for me is to primary, and keg it, and drink it. Now, of course I know that green beer isn't great beer- but I can't seem to help myself! So, if I keep it aging at room temperature for a bit, I can't drink it too early. Remember that carbonated beer is NOT conditioned beer- you still need some aging for most beers. In a bottle, you won't drink it too early because it'll be flat. In a keg, though, you can carb it up in 30 minutes if you have to! That doesn't mean the beer is close to its peak, though!