Do you take any knock back damage if you are on elevated and you fall down the transition ladder?

Yes, knock back is not movement so the elevation change is not used and you will be dealt 2 damage as per normal.

Thank you!

MisterID's answer is correct, but his reasoning is incorrect. Knock back is movement, just movement that isn't treated as movement by other game effects. However, ladders and stairs are irrelevant because the only question knockback off elevated terrain asks is did you cross the red line to a lower elevation. If the answer is yes, you end on the next square and take the damage. Nothing in the knock back section provides an exception for ladders or stairs, and nothing in the ladders/stairs section provides an exemption from knock back damage. Cross the red line to a lower elevation, that's it.

Anything else is trying to read real-world physics into the game. Which we all do from time to time. : - )

Nothing I said is wrong...

If this part was not in the Player's Guide, then they would not take damage since the description of Knock Back constantly uses the term movement.

It would seem that regardless of how you define Knock Back (movement, placement, whatever) if KB takes you from a higher to lower elevation level then you'll be dealt KB damage*: there's nothing in the rules I can see that would prevent that. Or have I misunderstood you?

*with the usual caveat for powers and abilities that specifically prevents the damage, like Flight or Combat Reflexes.