A bed is nothing more than a word that describes a comfortable (and, perhaps, safe) place to sleep. So...
Let's start with what we know
So What is Sleeping, Exactly?
Before we discuss fish sleep, lets figure build a platform by first deciding what sleep is.
The French psychologist, Henri Piéron laid down the definition of sleep in 1913 that is still used today:
– A stereotypic or species-specific sleep posture. We’re used to this in mammals. Most of them lay down in order to sleep.
– Maintenance of behavioral quiescence. In other words, the animal becomes inactive or dormant. It doesn’t eat or move around a bunch.
– Elevation of arousal threshold for stimulus. That is fancy scientist talk for the animal being less aware of its surroundings and not reacting as strongly to a stimulus.
For example, someone could walk into a room and say your name when you were awake, and you would immediately react. But, if you’re sleeping, someone might need to say your name several times, and loudly, for you to react at all.
– The state must be reversible. So, if you give an animal enough stimulation, it will wake up. This distinguishes sleep from things like being knocked unconscious or slipping into a coma.
No matter what kind of stimulus you give, someone in a coma will not simply wake up. So it’s different from just sleeping. (Source)
Summary: (a) There's a posture specific to your Merfolk. (b) They will be quiescent. (c) The could be less alert, but hold that thought. (d) And they can be awakened.
Next: do fish sleep?
The nature of fish "sleep" is an area of active research. While fish do not sleep in the same way that land mammals sleep, most fish do rest.
Research shows that fish may reduce their activity and metabolism while remaining alert to danger. Some fish float in place, some wedge themselves into a secure spot in the mud or coral, and some even locate a suitable nest. These periods of "suspended animation" may perform the same restorative functions as sleep does in people. (Source)
There's the issue with (c) above. Fish tend to remain more alert than mammals. But, to return to that first source:
Sleeping Posture: When fish sleep, they lie on the bottom, in a plant or hover almost motionless when they’re sleeping.
Dormancy: A sleeping fish will stop swimming, or swim very little, and stop eating.
Unresponsive: A sleeping fish won’t react to someone walking up to the tank, peering over the surface of the water, or food being dropped in the water.
You Can Wake Them Up: It is possible to wake a sleeping fish. Enough light or noise will eventually get them to wake up.
What changes might we care about with an intelligent species?
The semi-alert state that fish experience when asleep is to protect them from danger. Not just from predators (but that's certainly an issue), but also from stuff moving around. Currents carry things. Sometimes sharp pointy things like sticks. Granted, your merfolk might be deep enough that a stick isn't much of a threat, but they evolved, right? And that evolution would have had basic survival traits that would appear in current behavior.
Finally, let's return to my first statement and ask, what's a bed?
I've personally seen fish sleeping in stands of grass. But I've also seen a whole school of trout asleep in a shallow section of river, no more than an inch of water above them, swimming automatically to remain stationary in the water — and in the warm sunlight.
And that discrepancy invites a problem. You haven't explained how your merfolk react to sunlight. But let's set that aside. You do explain that they're city builders.
They'll prefer to rest indoors most of the time for privacy and protection of both possessions and family.
Being inside, currents are not longer an issue (at least not any more than human heating/air conditioning would cause).
They are in a medium that promotes 3D travel. That means they'll want to secure themselves. IMO, most intelligent creatures aren't fond of super-tight spaces, so I'm not inclined to believe they'll build closets to sleep in.
They will appreciate comfort, and comfort for a fish is warm and safe,so I'm thinking a warming source combined with natural plant life, which will hold them in place just fine. They won't float up because (a) fish don't do that when they sleep unless there's a current to move them and (b) there's a roof.
What's a bed?
A stand of plant life: grass, kelp, I can imagine pride would lead to a desire for rare plants to sleep in, but I can also imagine plant texture and the equivalent of underwater odor would also contribute to desirability. The plants will be periodically replaced, despite having been planted in the floor of the room, to keep them fresh (not unlike buying a new mattress).
Water conditioning: meaning freshness, oxygenation, salinity, mineral content, biological content (hey, they're merfolk, do they have gills? This could be a food source), and temperature.
The real question here is what they'd use as an alarm clock. Work starts at eight, after all.