Not really. H2O naturally dissociates into hydroxide and hydronium ions. The hydronium ion is formed when the oxygen from one water molecule “steals” a hydrogen atom from another, thus giving an ion with the formula H3O+. For simplicity, the hydronium ion is often suppressed in chemical equations and it’s unhydrated equivalent, the hydrogen ion H+ used instead. Even that, though is not a hydrogen atom. A hydrogen atom is a proton plus an electron. The hydrogen ion is a just a proton.
BTW this dissociation of water is a very weak reaction; almost all the water remains undissociated.
Isn’t it the equal number of H+ and OH- that results in a neutral pH for pure water?