Fly to each of the systems listed in the quest in order.
It doesn't have to be in order. There's three or four systems in a sector patrol, and as long as you visit all of them and complete their specific missions (like talking to the miners in the Beytan System), you'll have completed the sector patrol.
you should have a list some systems you need to patrol. You just go to those systems and you will get a mini mission for that system. Mostly shot some bad guys but sometimes it will be something else like talk to miners or fix some broken stuff.
i was in the vulcan sector and there was only just the planet and then i beamed down and there was no one to talk to or anything
when you are at the map where you can warp to sectors do you need to pick the individual section of vulcan vs just vulcan?
If you don't visually see the other systems listed in your patrol mission, hit M to open up the big map. There you have lots of dots with system labels. You can even click on one to plot a course to fly to it, or you can fly manually to it. The mission should list which systems in the sector you need to check out.