Welcome to FFXIV!

Unfortunately, duties are probably not the place to meet people. A lot (most?) of the time, people are just there to get the daily reward and just want to get through it as quickly as possible. Also, most duties have you teamed up with random people you will never meet again, so all around there is little incentive to talk.

Perhaps a better way to meet people is to hang around the Aetheryte and try to strike up conversations with people using the /say chat. Other people within earshot may also jump in. And if your server is like mine, Limsa Lominsa is the most populated out of the 3 starting cities so you may have better luck there.