No. The average ant can't. Worker ants are wingless.
However, both male ants and the virgin queen ants have wings that allow hem to fly for the mating. Once fertilized, the queen ant bites off her wings and starts the colony. The male ants, having done their duty, die.
So the answer becomes - most can't, but some can and do in the early summer.
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If you are asking about a queen honey bee, she can, but rarely does. She flies on a mating flight not long after she has hatched and then again when the colony decides to swarm. Apart from that, she remains in the hive or within the colony.
With their wings. You may be confused by the fact that most ants you see don't have wings, but all ants are members of the order Hymenoptera, which means "membrane wings." It's just that the sterile worker females are wingless; the queens and drones (which you normally don't see) do have wings, just like the related bees and wasps.
Yes, kind of.... but not proper queens of the nest. When they find a suitable nest they will land, bite off their wings and if they are accepted, they will be a PROPER queen of the nest.