They do pick up the pebbles in their mouths as every now and again when one swims to the top, it spits them out. I also had a few occasions where I go to take the lid off the black filter on the right to find there's a few pebbles on top of the lid.
With my previous 4' aquarium, I had a smaller filter pump with wider inlets. One cheeky koi often spat pebbles at the filter and occasionally managed to get a pebble or two inside. As you can imagine, this created a noisy racket with pebbles trapped inside the impeller and quite irritating when they do this in the middle of the night! I still could never figure out how the koi got the pebbles in there as any debris usually gets trapped in the filter's sponge. The only clue was when the filter flow rate slowed down when it needed cleaning as a koi probably could have tried forcing pebbles into the output.
What is interesting is that they don't cover the decoration boat in pebbles. There's usually a few pebbles randomly dotted here and there on the boat, but what I could try is putting a jug worth on top and seeing whether they lift them out.
In our goldfish and aquatic frog tanks, we never had any unusual pebble shifting, let along pebbles stuck inside the filter pump.