When Does the MonsterVerse "Godzilla: King of the Monsters" (2019) Take Place?

"Godzilla: King of the Monsters" is an adventure horror (kaiju) film written by Michael Dougherty with Zach Shields and Max Borenstein and released May 31st, 2019. 

It takes place on the fictional MonsterVerse Universe timeline in the year: 

2019 AD 

We know this because a TV news report very early in the movie, declares, "Godzilla, who hasn't been seen in over 5 years" presumably, since "the 2014 attacks" mentioned at the same time. This refers to the events of the previous film, the first canonical MonsterVerse film "Godzilla" (2014), the main action of which takes place in 2014 AD. 

2014 + 5 = 2019. 

Additionally, there is a 2014 AD prologue with a shot returning to the final events of the previous film and including a time and place setting title card that reads: "SAN FRANCISCO 2014". 

The end credits include close-ups of articles to give the audience bread crumbs for things that happen after the conclusion of the main action. This forms a kind of epilogue. One of them is on the website for a news channel "WPKW" that's dated for "Oct 28 2019". Another is on the BBC's site and is dated "Thu August 16, 2019". A third is for another TV station, "KNBS", and is dated "Oct 21, 2019". The final one is on the "ABC News" website and has the date: "June 21, 2019". 

There is also a post-credits scene that takes place an unspecified amount of time later than the main action and is therefore unplaceable other than taking place before the next film, "Godzilla vs. Kong" (2021). 

It is a chronologically midway film in the MonsterVerse Universe (a third chronological film to feature Godzilla) and can be watched at its timeline point.