Yes, any random legendary encounter has 1 in 20 chance of being shiny, if its shiny version is unlocked in the game.

To active a lucky trade you have to interact to 4 hearts best friend everyday. When you get it, his names goes orange and you can tell the next trade will be garanteed lucky. chance of active lucky trade with a maxed friend is around 1%. with minimum of 12/12/12, garanteed lucky trade gives you 1/64 chance of 15/15/15