If I remember correctly that’s kind of what happened. The Klingons used augment genes from Earth’s eugenics wars (the ones used to make Khan and others) to try and create their own augmented Klingons. It went wrong and the result was human looking Klingons and it somehow became a transmissible virus. That’s where the TOS looking looking ones came from. All of this is retcon of course.

Edit: What I wrote was pretty close. Here is a link to the Klingon page on memory alpha. The explanation is just after the start of the “History and Politics” section.