Does zebra fish need oxygen

Yes.

All fish need oxygen. Water is H2O, hydrogen and oxygen, and fish get the O when the water passes through their gills.
In an enclosed body of water like a fish tank, O can get depleted and that's one reason for changing the water periodically and having a bubbler (O in air too).

I have Zebra Danios in a tank with no air pump and they do just fine.

Of course air pumps and filters will prolong the life of your fish but there are some who do much better than others without.

Oxygen is dissolved in water, and that dissolved oxygen is what the fish take into their gills. Bubblers and the like help, but are not the main source of oxygen in an aquarium.

The surface area of the tank and the fish population, or rather the total length of all the fish in the tank, are the most important factors to maintaining sufficient oxygen in the tank. There is a limit to the 'inches of fish' an aquarium of a given size will safely support, and any aquarium book will give you this information.

Live plants in the tank, or anywhere for that matter, will also help since plant metabolism takes in carbon dioxide from the water or air, and gives off oxygen in exchange.

There are some fish that do not take oxygen from the water column. Bettas' and Gouramis' are labyrinth fish. The labyrinth is a breathing organ that is in the head just behind the gills. This structure allows the fish to take in atmospheric air. With this ability the fish can easily survive in waters with low dissolved oxygen.

They need to breathe air from the surface of the water.

They inhabit Africa, Asia,India, and Southeast Asia.