Does redshifting light due to cosmological expansion have anything to do with velocity

Say i set up an experiment. First i shine a lazer towards another galaxy where my friend setup a mirror. Say the total distance the light travels is 1 billion light years. I would expect to see the light to be redshifted through this journey. Now say my friend takes a beam of light in his spaceship the same distance. But his ship is traveling at 99.9999...% the speed of light. So the 1 billion light years only take 1 year to cross from his perspective. Would he see his light be red shifted the same as the light that went 1 billion light years in free space. I ask this because if it did he could measure his speed using the rate at which the light is redshifting inside his ship.