Both of nuclear fission reactions and fusion reactions result in the creation of new atoms, although of course these are produced from previously existing atoms.
The very first atoms, nearly all Hydrogen and Helium, came into being several hundred thousands of years after the big bang.
Before that the Universe was a very hot plasma of particles, too hot for electrons to bind with protons.
After that the first generation of stars came into being, these fusing the H and He into bigger atoms - other elements.