No, the moon will not turn green. Simply put, that’s an annual hoax that circulates social media, often as a meme. EarthSky has a full history of where this hoax started, as published by Snopes. In fact, for this April 20th, the moon is no where near full, as the memes portray it.
The moon can take on shades of yellow and orange and even red, naturally when it is low in the sky; when there is smoke are ash in the atmosphere; or in the process of an eclipse.
Some smartphone and other cameras may take a photo of a moon with a tint but that is a function of the electronics of the camera.
Moons of certain months have nicknames that are often related to folklore, culture or religion. They don’t often relate to what you see! Here’s more on a blue moon and other nicknames.