Can a faulty alternator trigger the Check Engine light?

I have a 2010 Toyota Corolla S. Couple of weeks ago while driving my battery light came on. I replace my alternator after that. That battery came on again after 2/3 days. I took my car to a different mechanic. He told me my alternator was undercharging my battery. I need to replace it. So, I replace my alternator again. Again, after running 2 days. My battery light came on and this time none of my lights (indicator/tail/head) was working. However, this issue only occurs after driving the car for around 30 minutes. I take it back to that mechanic and he told me he is not seeing anything unusual and ask me to take it to Toyota Dealer. I took my car to the Toyota dealer yesterday. Yesterday, they could not figure out what’s going on. This morning, they called and tell me that after diagnosis (I have to pay $375 for diagnosis), they are pretty sure that my problem is coming from the alternator. So, I asked them to change it again for me. I just want to get done with it. About 20 minutes ago, I get a call from them, saying that, they changed the alternator and after running the car for around 2.5 hours, my check engine light and battery light came on. Now, I never had my check engine light on. And even with the battery light on my car was running ok. They tell me that they will try to investigate further tomorrow and let me know. I was wondering what might cause to turn on my check engine light?

Moreover, I’m scared what if they try to charge me extra money for that (Even though, I never had my check engine light on ). This problem occurs during they try to fix my car, and in the morning they told me everything but the alternator is running as expected. Any tips on how can I deal with them in case they try to charge me more for fixing the check engine light or any further issues?