No, you don’t need an internet connection for using Bluetooth. Bluetooth and internet connection are two separate things.
The Bluetooth connection is limited to your surroundings. You can transfer things with Bluetooth connection if you are in a radius of 10m of another device. While with an internet connection you can send the file to the other side of the world 🙂
Also for working with Bluetooth, you don’t need to turn WIFI or mobile data on your mobile phone the data count for every connection is separated.