Yes.

There are no limits* to how high a pixie may fly, provided that they end their turn only one square above the ground. However, if they do end their turn above their altitude limit they fall with the consequences laid out for falling (1d10 damage/10 foot of drop).

*The theoretical maximum height a pixie may reach on their turn (with no extra movement trickery) is 8*3 + 1 or 25 squares high (run 8, trade standard for another run 8, action point for another run 8, +1 for starting height), that's 125 ft at the end of which they would fall. Falling says to subtract the creature's fly speed so that leaves us at 95 feet of fall, meaning 9d10 falling damage.

Flying creatures can remain aloft unless their rules says otherwise

The 2019 Sage Advice Compendium contains the following paragraph which directly addresses this situation:

Can a flying creature without the hover trait stay in one place while airborne, or does it need to move each round? A flyer that lacks the hover trait can stay aloft without moving each round.

From the Flying Movement section under Movement and Position (PHB, p. 191) we have (emphasis mine):

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.

Possessing the 'hover' ability prevents a creature from falling when knocked prone or deprived of movement, but otherwise has no effect on where the creature may end its turn. By default creatures that possess a fly speed may end their turn in the air. However, doing so without the 'hover' ability exposes them to falling damage should an enemy somehow deprive them of movement.

Some creatures or abilities may limit your flying ability and require you to land at the end of your turn. See Do I have to land at the end of my turn? for evidence that this is not the general case. One example of a feature with this restriction is the level 14 Path of the Totem Warrior (Eagle) feature which says (emphasis mine):

While raging, you have a flying speed equal to your current walking speed. This benefit works only in short bursts; you fall if you end your turn in the air and nothing else is holding you aloft.

If the feature allowing you to fly does not possess text to a similar effect, you can end your turn in the air or remain stationary while aloft. Having the 'hover' ability merely negates the risk of falling.