You only need an "Umwelt Plakette" if you travel into an environmental zone, which a pretty much limited to cities. They will check foreign plated cars and it is enforced.
Lots of information here,
https://www.umwelt-plakette.de/en/i...es-in-germany/german-environmental-zones.html
So for the Autobahn you wouldn't need one, BUT if you end up going into a town that's an "Umwelt Zone" you'll need one, and pretty much all are now GREEN only (Euro 4 or better).
They are issued by the Tüv and you can buy them online. Costs €15 I think and I think you can get them through ADAC/RAC etc as well but be careful of 3rd party sellers.
For foreign registered cars its based on the age of the vehicle so if it was first registered before 2006ish you probably wouldn't get one.
As for a quick route to Turkey? If you're avoiding Austria for the tolls, you'll need to avoid Slovenia as well (as you'll need a Vignette to drive on Slovenian motorways and Croatia's motorways are toll roads as well. You may find cheap means A and B roads for much of Europe South and East of Germany.
Oh and ffs don't run out of fuel on the German autobahn, its an offence to do so.