They did, though the purpose was as much propaganda and antisemitism as it was about protecting animals.
In 1933, the Nazis passed the Reichstierschutzgesetz (animal protection law of the Reich), which punished those that 'tortured animals or caused them long or repeated, substantial pain or suffering'.
According to historian Mieke Roscher, the goal was to exclude Jews and Roma from German society. Previous laws already criminalized shechita, and the Reichstierschutzgesetz was specifically used to remove Jews from sciences which performed animal testing. It was also used to target Roma circuses. Other circuses as well as animal testing for war purposes or by some non-Jewish scientists on the other hand were not persecuted. Not all animals were treated equally, wild and pure-raced animals were protected, while pets and vermin were not.
The preamble of the law already shows that the law is based on a German-völkisch ideology, when it references the German Volk and its moral obligations.