Well I'm glad you went with the 135mm f/2L. I own both lenses you were asking about but I SPECIFICALLY take the 135mm f/2L with me when I do concerts -- it's fantastic for that. The "other" lens I bring is the 70-200mm f/2.8L IS. If I need to get shots of people after the concert where I'll be much closer then I switch to the 24-70mm f/2.8L. That's pretty much all I use for concerts.
Lighting will be HIGHLY variable. So there's no "right" answer. Fortunately you can crank the ISO on a 5D III when you really need to.
The switch on the side (.9m - ∞ or 1.6m - ∞

represents the focusing range. 1.6 meters is just a bit over 5'. In the 1.6m - ∞ mode the lens will not bother to hunt for focus on anything closer than 1.6m (5 1/4') away. You can use that mode for concerts since it's unlikely you'd ever be shooting anything closer.
You can use shutter speed either to freeze action or to deliberately show motion.
Here's an example of a concert shot with the EF 135mm f/2L ... although this one was taken with my 5D II body (before I owned a 5D III).
At typical sizes you don't notice the noise (this image has deliberately NOT been de-noised... I often use it as a reference because it was taken at ISO 6400). There is noise when you got to 100% crop.
Chris Koza in Concert by
Tim Campbell1, on Flickr
This one happened to be taken at 1/60th. You can see blur in his hands, but not in his body or face. But as I'm at ISO 6400 and f/2 in this lighting, I'd have to go to ISO 25,600 to get the shutter speed up to 1/250th. I don't mind a little motion blur to imply action. Imagine if I'd have only had an f/2.8 lens... I'd be down at 1/30th to get this. Using a kit lens with variable aperture up to f/5.6 would just be a non-starter... you'd be stuck at 1/8th sec. in this light.
Always try to capture the energy of the event. Shots aren't nearly as interesting if it looks like there's just some guy standing at a microphone.