The spell-check application of most word processing software programs would not catch a slip-up of these two words. Spell-check is looking for words that aren’t in its dictionary, and words that resemble words in its dictionary but are possibly spelled wrong. Spell-check isn’t perfect. It doesn’t know and can’t guess what word you wanted or what word you meant, it can only judge the words on the page. If you used words that are all spelled correctly, it gives you a pass anyway.
Autocorrect suggests words that start with the same letters. It’s suggesting what word you may want to save time, but quite often, its suggestions are pretty off base. They don’t help you out, but they do make you laugh.
Pidgin is a noun. It is a language that develops as a hybrid of other, more official, languages.
Pigeon is a noun. It’s a type of bird with short legs, small heads, and large round bodies. Pigeons tend to bob their heads as they walk to maintain balance. Pigeons frequently live in cities where there’s plenty of human food scraps to eat, and lots of nooks and crannies to build nests in. (Doves are similar to pigeons, but they smaller.)
The following story uses both words correctly:
Plegmund had a kit of pet pigeons whom he could speak to via coos and whistles. He also had a gift for languages, and understanding obscure pidgins of those languages.