As far as I'm aware, that particular phrase was in regular conversational use before The Matrix, but it's popularity as a meme is obviously because of the movie. It was not a "catch phrase" for any other character or work that I know of prior to Morpheus.
So yes, the phrase and the meme did come from The Matrix. It's just a mis-quote. It's extremely common for popular phrases from movies and TV to get altered slightly as people repeat them and misquote them. This is particularly true when the mis-quoted version is more pithy or more succinct than the original.
In The Matrix, Morpheus gives a long speech about the Matrix, which starts off with "Let me tell you...". He then proceeds to tell Neo some very unbelievable things. Later, in the Construct, he says a number of similar lines of dialoge of the general sense of "Why do you think xxx is real?" Most likely, the all of this dialogue has been compressed to it's basic concept, expressed in the far more meme-worthy, "What if I told you..."
There are a number of other famous examples. For example:
There's not one time in any TOS show or movie where Kirk says "Beam me up, Scotty!". In most cases, he says some variation of "Enterprise, [1/2/3] to beam up", and usually only mentions Scotty by name when things are going bad. There are times when he comes close, but it's never exactly those words. (Shatner did eventually use the phrase exactly as commonly quoted, but it was long after it became a pop-culture phenomenon, as an homage.)
The most famous line in perhaps all of Star Wars is "Luke, I am your father". But Vader never says that. The actual dialogue line, in response to Luke's "He told me you killed him", is "No, I am your father."
There's no point in Casablanca where Rick, or anyone else, says "Play it again, Sam." The actual dialogue is several lines long.
and there are others.