Is it true that once we reach adulthood we never produce new neurons?

The short answer is no, this is a false statemement. There are two areas in the brain which undergo adult neurogenesis after adolescence. These is the subventricular zone of the lateral ventricles, which supplies your olfactory (smelling) system with new interneurons. And the subgranular zone of the dentage gyrus which generates a particular type of neuron called a granule cell. You might have already known that, but did you know how contraversial adult neurogenesis is in the world of neuroscience? Last year two papers were released directly contradicting one another. The first paper titled 'Adult neurogenesis is overrated' argued how little a contribution it made to the adult brain. The second paper disputed the methods and findings of the first. When you start to delve into science and ask more complicated questions, the answers are rarely simple. However, the joy of science is engaging and exploring down avenues you didn't know previously existed.