The simple answer is no, dolphins don’t have gills.
Gills are actually a characteristic of fish, sharks and a variety of amphibious creatures that enables them to pull oxygen directly out of the water so that they can breathe in an aquatic environment.
Depending on the species some amphibians can breathe both underwater and on the surface, however most fish and sharks can only breathe underwater and would actually suffocate above the surface of the water.
In contrast to fish and sharks dolphins are marine mammals and must come to the surface in order to breathe.
Dolphins are equipped with lungs (like humans and land animals) not gills and would drown if they tried to breathe underwater just as a human would drown if they tried to inhale while submerged underwater.
In order to breathe dolphins swim to the surface and inhale oxygen through their blowhole.
One interesting characteristic that separates dolphins from other mammals is that unlike humans and land animals dolphins are incapable of breathing through their mouth because their trachea and air passage are not connected.
This serves several purposes for dolphins.
First having a separate trachea and esophagus allows dolphins to surface partially above the water while only exposing their blowhole, which makes it easier to breathe and allows them to rest without having to constantly monitor whether or not their blowhole is above the surface.
By having a blowhole at the top of their head it allows dolphins to breathe more easily during resting periods.
Secondly, by having a separate breathing system dolphins are able to hunt for and consume fish and various other prey while under the water without the fear of drowning.
They don’t have to be concerned with opening up their lungs and accidentally filling them with water while consuming fish and other prey.
Some dolphins such as the killer whale may be seen spouting water from its blowhole when it rises to the surface of the water.
When this occurs the dolphin is simply blowing the water that has collected around its blowhole away.
It isn’t water that has been ingested through the mouth, lungs or blowhole, but water surrounding the outside of the blowhole that has been powerfully forced away when the dolphin exhales.
Dolphins like humans would drown if water entered into their lungs.