Peaches do in fact contain cyanide, though it is a very small amount. You would have to eat quite a few peaches in one sitting to make yourself sick and it is quite difficult to do so. Bitter almonds also contain a small amount of cyanide which is enough to make a person sick if they eat enough of them.
Sometimes people accidentally eat the pits from peaches and apricots, though in order for them to actually get sick from the cyanide found in them, they would have to eat more than most people do. In the last half century only nine people have died from cyanide poisoning due to peach or apricot pits, so it is a rare occurrence though it does happen every so often.