The answer is almost certainly yes - I’m not familiar with openmaptiles.org but their site includes a “you can edit them” page: https://openmaptiles.org/docs/style/maputnik/ .

It’s worth mentioning though that openmaptiles.org is only one of many providers - the maps you see on openstreetmap.org are generated by a different technology, and many, many different styles are available - either self-hosted or hosted by someone else (usually for a fee if you’re making lots of requests).

What does your existing Google Maps plugin do, and where in the world are you interested in displaying maps for?

There are a lot of links here https://wiki.openstreetmap.org/wiki/Category:Slippy_map . Many options use Leaflet https://wiki.openstreetmap.org/wiki/Leaflet as a Javascript library; and if you want to create your own tiles using the same technology as used at OSM.org have a look at https://switch2osm.org/manually-building-a-tile-server-16-04-2-lts/ .