No, if I remember correctly the most probable fusion path are deuterium + tritium and deuterium + deuterium. In both case there are only 2 hydrogen isotopes fusing together, not 4.

Fusing 4 protons into helium definitely happens in the sun, where protons are the most common hydrogen isotope. Most fusion on Earth is done using the deuterium-tritium reaction, which produces helium and a neutron. The reason is that it's an easier reaction to do: Obtaining the special hydrogen isotopes is much less of a pain than trying to design a reactor that uses the 4 proton reaction.