Flying creatures need to make checks in order to hover, even when they have a natural flight speed (but they're often so good at those checks that they can automatically succeed).
As the general description of flying in the Fly skill states:
Without making a check, a flying creature can remain flying at the end of its turn so long as it moves a distance greater than half its speed. ... Taking any action that violates these rules requires a Fly check.
Unless otherwise stated in the creature's own description, a flying creature that wants to hover needs to make a Fly check in order to do. It doesn't have to hover while it's fighting; as long as it moves at least half its speed with a move action in every round, it's not done anything that requires it to actually make a check. However, it will often be beneficial to hover in place, to allow for using full-round actions, avoid provoking attacks of opportunity, and so on.
Many naturally flying creatures have Fly modifiers which are so high that they will always pass most flying checks in normal circumstances - remembering that skill checks, unlike attack rolls and saving throws, are not subject to critical success/failure rules. A +14 or greater modifier is enough to ensure that a creature will always pass the check to hover in normal conditions, at which point there is no sense in rolling it. (Outside of combat situations, a flying creature that is not under particular stress can take 10 on their flight checks as normal for other skills, so it's even easier for them to make their checks.)
The quasit demon from your example has a +20 modifier (the bonus from its maneuverability rating and size is included in the listed modifier in its stats) and so it can easily hover without needing to actually roll flight checks unless it's in very strong winds or something like that.
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