A big assumption
Your question assumes that a dragon can choose to go prone while flying. I am not sure that is the case (emphasis and explication mine):
Combatants often find themselves lying on the ground, either because they are knocked down or because they throw themselves down. In the game, they [that is, those combatants who find themselves lying on the ground] are prone.
This passage could be interpreted to mean that throwing oneself down to become prone requires that one end by lying on the ground, and as such, cannot be done while still airborne.
Looking at the rules for flying movement,
Flying creatures enjoy many benefits of mobility, but they must also deal with the danger of falling. If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.
While these rules establish that a flying creature may be knocked prone, they do not establish that a flying creature can choose to 'throw themselves down' and go prone.
Thus it is a rather big assumption on your part that a flying creature can voluntarily go prone, when in fact that merits its own stacks question. However, for the sake of answering this question, let's assume that your dragon can indeed go prone while flying without having to lay on the ground.
Will it fall?
As quoted above, there are three situations in which a flying creature will be forced to fall:
(a) if it is knocked prone
(b) if its speed is reduced to 0
(c) if it is 'otherwise' deprived of the ability to move (here 'otherwise' is ambiguous, but since being knocked prone does not deprive one of the ability to move, we can understand this to mean being deprived of the ability to move in any way other than having one's speed reduced to zero)
Since choosing to go prone fulfils none of these conditions, a creature that chooses to go prone while flying (if permitted to do so) will not fall.
If it doesn't fall, what happens RAW?
Suppose we now have a prone, still-flying, not-falling creature.
First, they should have all the mechanical effects of the prone condition:
A prone creature's only movement option is to crawl, unless it stands up and thereby ends the condition.
The creature has disadvantage on attack rolls.
An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.
Further,
To move while prone, you must crawl or use magic such as teleportation. Every foot of movement while crawling costs 1 extra foot.
So we have a still-flying creature that cannot move as fast but is harder to hit at a distance, and that is easier to hit within 5 feet but which has a harder time itself hitting.
What I would do
How should we represent the rules consequences of flying while prone within the narrative of the game?
Being knocked prone is one thing, but choosing to go prone typically occurs when a character is protecting themselves from missile fire. To me, it makes the most narrative sense to leave behind the description of prone as being 'crawling on the ground' and instead assume that the flying creature is taking evasive maneuvers; rolls, high-speed turns, sudden changes of trajectory, and such. These actions would be under their own control, would reduce the net distance they traveled, would make them harder to hit unless their attacker managed to get in close, and would make it more difficult for them to attack, neatly fitting all of the mechanical effects of being prone while flying.
Thus, narratively, I would assume that:
The prone condition on the ground means that you are crawling
Being knocked prone while flying means that you are falling
Choosing to go prone while flying means that you are making evasive maneuvers
This is my opinion; other narrative interpretations may apply in other games. However, the mechanical consequences of being prone while flying (assuming one can choose to do so) are RAW and must apply regardless.
