Yes, they do!
Although it may look considerable different from humans or fluffy pets sleeping, most fish require rest periods where they reduce their activity and metabolism. Many fish must stay alert to danger in this low-energy state, so there is still some brain activity. Fish do not have eyelids, so they cannot close off their sight. Those individuals undergoing migration, spawning or caring for young may hold off on sleep until that period has passed.
Depending on where a fish lives and its activity level, a fish may simply settle to the bottom of their watery home, secure themselves in a crevice like a triggerfish, float in place (thanks to neutral buoyancy) or settle into a nest. Ram ventilating sharks have to keep swimming in order to keep breathing, so they are able to hit this low power mode and keep swimming!
So, although it is not “traditional” sleep, fish DO enter a conservative energy state.
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