“I've tried to suppress it.”
Your friend is from the southern part of the U.S., but he doesn't have a strong Southern accent. He explained to you that he used to have one when he was younger, but says this to explain what happened to it.
I've tried to suppress it.
"Suppressing" an accent means hiding your natural way of speaking. It may help to learn that the word "suppress" means to stop something or hold it back. It's used in these phrases:
suppress a cough
suppress a smile
suppress your appetite
suppress an urge (to do something)
You usually can't suppress a foreign accent. For example, if you're from France and learning English, you have a French accent. You might be able to improve or correct your accent, but you can't suppress it. But some people can suppress their regional accent, like a Southern accent.