No, 4-track tape doesn't signify Quad. A 4-track open reel tape signifies that there's 2 channels x 2 programs. In this case, it means a 2ch tape that has a forward and a reverse direction.
A Quad reel is still 4 tracks, all running in the same direction. It will be called Quad, or 4-channel.
There were also 4-track cartridges, the predecessor to 8-track cartridges. These had 2 programs, each with a stereo pair. In this case, both programs read in the same direction.
An 8-track cartridge is not 8 channels. It is either 2ch x 4 programs or 4ch x 2 programs.