No. Bluetooth, WiFi and Internet connection are three totally different interfaces.
Bluetooth normally provides a link between either two computing devices (PC, phone, tablet, etc) or a computing device and a peripheral (headphones, speaker, keyboard, mouse, etc).
WiFi is normally an interface between a router or wireless access point and a device such as a PC, phone tablet, TV, printer, etc. Some people also use WiFi between a router and a games console (PS4, Xbox, etc) although this is generally a bad idea as WiFi performance can be erratic due to other users and interference from other networks, which can create problems for many online games. Gaming devices almost invariably do better with Ethernet connections.
Ethernet provides an alternative interface to WiFi for connecting devices to a router.
There is a mode of WiFi (Ad-hoc network) that allows two computers to communicate directly without a router.
The Internet connection is on the other side of the router and provides the connection to the Internet Service Provider (ISP) and the Internet itself. Bluetooth, WiFi and Ethernet do not come across the ISP connection but are local interfaces within your own home.
You can have Bluetooth, WiFi or Ethernet connections (and connections between WiFi and Ethernet) without any Internet connection to allow local private networking between devices.
Finally, I am not aware of any router that supports Bluetooth. No WiFi or router (or Internet connection) is required to use Bluetooth.
I hope this helps.
No, bluetooth is usually a "device to device" short range connection. Like a cell phone to a headset, or TV to a speaker, or computer to a printer. It is way to slow for internet use unless there is no other way to connect. Wifi and Bluetooth are two different technologies. Neither depends on the other.
Bluetooth is independent from wifi.