Yeah, I also don't think I show might happen (Though due to the gap between the two films and Godzilla's history with movie TV Shows chances) are much higher than with other movies
An idea I came up with is that the series's premise in the first season: (Remember that series always have different plotlines going on at the same time throughout all the episodes, unless it's one of those "slice-of-life" style shows in which every episode is self-contained).
NOTE: I DIDN'T DO ANY GRAMMAR CORRECTIONS DO THIS AND PROOFREADING because I was in a hurrry THERE WILL BE SOME WIERD WORDS, MISPELLINGS AND LOTS OF OTHER STUFF (I will fix it all later)
A batch of MUTO eggs have survived the explosion (sounds like Godzilla: The Series, huh?), however, that is not known to the audience or the the characters, and think that only one Muto has survived. Most of the show follows a new cast of characters who are part of the MONARCH agency (Ford Brody could be an awesome cameo in a special episode) One of them a conspiracy theorist who used to work with Joe Brody (The the MONARCH needs now after a whole lot of new monsters have started appearing), There also seems to be a spy in the midst of them (Which they discover thanks to data recordings) but it us unknown if this spy is american or japanese (For those who did not read Godzilla: Awakening, MONARCH are joint american/japanese). The male MUTO, who is still in his infant stages (right now about the size of a dog) is looking for it's mother, who's body (and decorticated head) lie in the MONARCH base. The MUTO, nicknamed "Darkie" (bad name right?) due to weird black spots found on him by a witness (the black spots are the effect of some damage caused by the fire Ford Brody caused in the movie). MONARCH is very busy, with the Darkie to track down, a spy, and a new batch of monsters aweakning after the bomb at San Francisco. At some point after the MUTO has become some sort of a public enemy, sometimes attacking shops, people's houses and the streets in search of radiation. Now taller then a humane male (imagine as tall as the tallest person in the world) the MUTO attacks the very secret MONARCH base while most of the forces are dispatched to check on a tunneling monster in India, leaving the base lightly defended. The MUTO was able to track his mother's scent all the way to where her body is in the base, after fatally wounding one of the characters (We need some deaths) and nearly killing another, he sucks up all the radiation from his mother's corpse, making him grow much faster. The MUTO is revealed to be a winged Male, and is now as large as the pterosaurs in Jurassic Park III, and flies away. Darkie has now time to waste, and flies back to San Francisco (still being repaired and still with radiation signs from the bomb at the end of Godzilla), he calls upon all the Mutos and it is now revealed that a whole lot of MUTOs have survived. What happens next is a big war between the juvenile MUTOs (who are much, much smaller then what we seen in the movie) and the Military. At some point, Godzilla comes to stop all of this and a fight begins between him and all the other MUTOs. Meanwhile, Darkie is killed by one of the main characters, but that is right after he just nearly killed another one of the main characters. Main Char 1 must carry Main Char 2 in the midst of the battle to a medical bay, while Godzilla fights the MUTOs who are now using an ability they did not use in the movies (but used in Godzilla Awakening). MUTO cells are able to move away from each other to change the MUTO's form and looks, and even split to become smaller entities or join to become bigger, this all happened in the GODZILLA AWAKENING. The MUTOs, now mature enough to be able to movie cells, clump up and become one big giant MUTO, Godzilla has an awesome fight and ends up disabling the MUTO Cells with his atomic breath (Like in the comic, Godzilla Atomic Breath is the only thing that can disable MUTO cells) and kills the Mega-Muto by firing a concentrated blow of his breath right through it, making a big gaping hole in the MUTO's heart, killing it.
Okay..that was a wall of text...
?Buy anyways, that was just the first plotline, Godzilla will be constantly appearing whenever MONARCH can't handle some of the monster it awakens. A monster fight between Godzilla and some wicked sea creature accidentally cause Japanese and American forces to collide, *accidentally* causing WWIII to begin (The whole situation turns out to be caused by human spies, who's benefactors will benefit from the war). Godzilla finds himself in the middle of the war, the Japanese think that Godzilla is working with the 'muricans, and the americans blame Japan for Godzilla. Meanwhile, MONARCH crumbles as it is governed by both Japanese and Americans, and things start to fall down. The spy in the MONARCH turns out to be some really nice character we got attached to, and is american, and a military attack occours on the MONARCH base, whose location has been discovered thanks our old spy friend. During the attack, the spy steals everything from the database and the MONARCH base is destroyed (The Military also decides to kill all witnesses) leaving our main characters *plus some other guys* as the last of the group. It is revealed that the americans intend to use Monarch's research to attempt to control the monsters and destroy Japan, but Japan wants to do the same. Our ragtag group of misfits awaken Godzilla. The american controlled Monsters go mad, so do the Japanese, and an all-out monster war happens. Godzilla intervenes and decides to fight them all, and it is revealed that his whole war was planned by aliens (Xilliens!!) who wanted Earth's inhabitants to destroy themselves so they can colonize it. The Xilliens set up a giant beacon in the middle of Abu Dhabi (we want to go international) and all of the monsters are attracted to it. Soon, the monsters start fighting over the beacon, but Godzilla comes and beats them all and defeats the Xillien's trump card, and the war ends. The americans and Japanese come to good terms, and the Xilliens retreat, Godzilla is too strong of an adversary.
?That was just some mad speculation.