In daily speech you can or would produce "a document" or "documents" for inspection. So you can produce an envelope ( letter presumably) from your pocket or an envelope with a document in it.
To take or pull anything else from your pocket you would simply take out from your pocket, or informally "pull it/something" out of you pocket.
In the UK you may be asked to "produce" driving documents to your local police station by the police, or other government departments.
So it is has Jacque has explained but for taking things from your pocket it is generally to produce/show documents to official people and departments. For anything else it is "take from your pocket" . It is unfortunate that the definition of to show is not very well explained in many dictionaries, the "bring forth" definition is probably confusing to manyforeign learners.