If you want to get a 256*256 PNG tile of a certain map service from geoserver you can do it with including tiled=TRUE in your request:
But there's no zoom parameter in the request, and the client side application will take care of the certain zoom level and the bounding box.
But if you insist on using WMTS (which capability is included in GeoServer with GeoWebCache integration) you can do it with the following request:
Here the TILEROW and TILECOL are the key parameters. But if you want to extend your knowledge about WMTS I would suggest you to check out the official OGC standard here:
And you can find useful information about WMTS in geoserver at the following link:
I hope this helps!