Is there a way to have multiple hives or something to increase the egg production ?

Yeah, there is. I'll spoil it down here so don't read if you want to discover it by yourself!

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To make Nutritious Eggs a hive uses Berries. When a hive stocks up 4 Nutritious Eggs, it gives birth to a new Queen. That Queen will look for another Twig Bush to start a new Hive. Don't forget you can plant Twig Bushes and Berry Bushes on your own Farm Plots to help with all of this.

Thank, thats not something i could have guess maybe you should make a way in game to learn about it if not already done.