Could desalination plants filter out gold and silver from salt water?

In a reverse osmosis (RO) desalination plant the salt and the gold and silver will be concentrated in the retentate.  They are still pretty dilute though.

Absolutely. Its just that the return on investment is too low with current technology to make it worth while. If someone could invent a type of filter process that would not cost so much they’d be sitting on a gold mine. And a silver mine too. Literally. 

Sort of. The cheapest method is solar distillation, which removes fresh water for drinking. What is left is a brine, which contains a lot of salt and trace amounts of precious methods. Allow the water to evaporate and you get mostly sea-salt, with trace "contamination" of gold and silver. It would not yield enough of those metals to pay for their extraction.

Doodly doo doo doo....Perhaps.

In a reverse osmosis (RO) desalination plant the salt and the gold and silver will be concentrated in the retentate.  They are still pretty dilute though.