According to Trek lore Vulcan is supposed to orbit 40 Eridani, which is known to be about 16 light years from our solar system. Covering 16 light years in "five minutes" would require a speed of 1,681,920 times the speed of light. I don't think any warp scale ever used in the Trek universe goes that high.
One way or another, either in depicting too high a speed or failing to depict the passage of time correctly, they goofed.