Telling where I am originally from at the first lecture?

You can certainly do so. Something like "I was born here. I did my undergraduate degree here, my masters degree here, and my PhD here. After my PhD ..." etc. It doesn't have to be very long and could also be part of your lecture slides.

One of my lecturers from my bachelor's degree did this, and I still remember she and my brother shared an alma mater.

I think it is a good thing to do, actually. In fact, anything you can do or say to make connections with them is good. In this case it places you in a context. You can also ask, assuming that the scale is reasonable, if anyone else comes from an "interesting" place. Don't press it if no one volunteers, but many people are proud of their origins. Others may not want to seem different from their peers.

The basic idea is to make yourself seem more human to them rather than just someone who will need to judge them eventually. Students sometimes fear their professors, especially in early courses when they have little experience. When they see you as a person they will probably be more willing to accept what you say when you, inevitably, correct them.

I usually introduce myself in my first lecture by giving a short overview about my background. Since I'm not very different from most students, there is nothing surprising in this, but if I would have a different background, I would mention it since it helps building a relationship to your students. If you are having an accent, they will recognize that you are not a native speaker anyway ;-).