So I was playing around with some different combinations on the Kuat Refit and a combo poked its head out at me. If I use the Phylon Q7 Tractor Beams on a ship without a nav token and it has to drop its speed say from 2 to 1, does it trigger the The Grand Inquisitor card so the ship he is on can change its speed if it is in range?

Yes, as it is the ship itself that is changing speed.

As long as it's an enemy ship that changed speeds and its distance 1-5 you can use the Grand Inquisitor

Card Text:

"When an enemy ship at distance 1-5 changes its speed, you may exhaust this card to increase or decrease your speed by 1."

It won't trigger on Konstantine though because he changes the speed instead of making your opponent do it.