Usually I find barbers don’t even try talking to me because they tend to be older and their clients tend to be older so they don’t know what to say. And I look a lot younger than I am. Not that I want to talk to them anyway, due to awkwardness, but seeing all the other barbers talking to their clients and mine not even trying to talk to me somehow makes it even worse
My barber always leaves a TV running in the corner. If you're in a talkative mood he's more than happy to chat, but if you're the quiet sort he's content to work quietly and listen to the TV instead. (I'm not a huge fan of TV, but it's prompted some interesting conversations about motor sports, of which he's a fan but which I know fairly little about.)
Before I found my current barber, I used to go to a 'hair salon' sort of place that had very talkative hip young hairdressers. They were always friendly but I found them harder to relate to. It was also a lot more crowded which didn't help, and they always seemed a little disappointed when I asked for a regular haircut.