Xanatar's Guide to Everything confirms that falling does not cost any movement.
The rules for falling are
A fall from a great height is one of the most common hazards facing an adventurer. At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.
and they do not state that you spend your movement (or part of it) for the falling: V2blast's answer covers this aspect in great detail.
Moreover, this is confirmed also by the XGtE:
The rule for falling assumes that a creature immediately drops the entire distance when it falls.
Since a creature immediately drops the entire distance, this implies that no movement is used to cover this distance, even if it is greater than a character's speed.
Be aware that the above quote, even if it is included into an optional rule, refers to Basic Rules for falling.
XGtE provides also some optional rules for falling from great heights and for flying creatures. The second optional rule (Flying Creatures and Falling) could be of some use if the character has a flying speed.
Rate of Falling
When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn. This process continues until the fall ends, either because you hit the ground or the fall is otherwise halted.
The above snippet confirms that one does not have to spend their movement while falling, otherwise one cannot fall for 500 feet.
Flying Creatures and Falling
If you’d like a flying creature to have a better chance of surviving a fall than a non-flying creature does, use this rule: subtract the creature’s current flying speed from the distance it fell before calculating falling damage. This rule is helpful to a flier that is knocked prone but is still conscious and has a current flying speed that is greater than 0 feet. The rule is designed to simulate the creature flapping its wings furiously or taking similar measures to slow the velocity of its fall.
In this case, it is specified that the movement can be spent for slowing the falling, for diminishing the damage.
The second part of this optional rule says:
If you use the rule for rate of falling in the previous section, a flying creature descends 500 feet on the turn when it falls, just as other creatures do. But if that creature starts any of its later turns still falling and is prone, it can halt the fall on its turn by spending half its flying speed to counter the prone condition (as if it were standing up in midair).
This hence allows to use half of the flying speed for stopping the fall. A strict reading suggests that you can use this strategy in two successive rounds in which one is falling from a height greater that 500 feet, but a DM may allow to use half of the movement to stop the falling also for smaller heights.