Keep in mind that the water in the Gulf is brown because of the Mississippi River stiring things up. So the further away you get from the mouth of the river the clearer the water will get. As to how far out............It varies on a day to day basis. Some days are worse than others.
But if I were to put a guess to it, I would say 20 to 30 miles from Galveston, you will start seeing blue water. I have been on fishing trips out of Galveston and was fishing in clear water, so it can't be that far from port.
Depending on what time of year you are cruising, it may be dark before you get into clear water...
My husband had went deep sea fishing years ago in Gulf and he remembered that it changed but couldn't remember just how far out they were in boat. He just wanted me to take notice of the change in color and couldn't remember a distance to look for.
Well, it doesn't change super fast - green then instantly blue... but it does change. Like Rusty pointed out, this is because of the shallow water and the Mississippi river and other rivers close to land that cloud up the water.
We went on the Conquest this last March. A great trip. Loved it all. There is beautiful blue green water in our Gulf. Florida doesn't have the brown color that Texas does. It's those nasty oil wells that mess up the Tx. coast so bad. I'm sure the Mighty Miss does add to the muck because of the gulfstream. Anyway you will love the Conquest.