No, a creature cannot use its fly speed while prone
The creature is lying down. However, if the creature is climbing or flying, it falls.
The only way the creature can move is by crawling, teleporting, or being pulled, pushed, or slid. (Rules Compendium 232, but the DMG says the same thing in different words on page 47)
So when a creature is prone, it cannot fly because that is not one of the allowed movement types. If it somehow managed to try, being prone makes a flying creature fall.
hover quality doesn't change this, because hovering only changes how flight interacts with the
Hover: A creature that can hover, such as a beholder, can remain in the air even when it is stunned. (Rules Compendium 210, rephrased on 312)
Please note that some of this is changed from the original mechanics; the PHB1 says that hover allows a creature to remain in the air without moving, for example, which is now just a thing anyone who flies can do.
All of this is by the book, not by common sense (which could be argued ad nauseam), because D&D 4e is strictly a mechanical construct and sacrifices simulation or reality-based adjudication for a homogenous structure that hopes to improve balance through uniformity. Physics need not apply, and what we "don't have any problem imagining" would be a house rule.
[All of these rules are pulled from the online DDI Compendium, Glossary section, which is kept updated with the most recent errata and provides page references for the books it quotes from. Unfortunately it requires a paid subscription to access. If you want a hard book, the Rules Compendium is the most up to date re-statement of rules as errata'd.]