In order to unlock the Insurrection quest chain, you first need to complete the Good Suramaritan achievement. (Which requires pretty much all the quesst in the Suramar zone.)
So, just keep at it. And be aware that you’ll need to do most of the zone’s side quests as well. You’ll also need to do the Leyline Bling achievement as that’s required to progress later parts of Good Suramaritan. So, that’s another achievement to keep track of.
Basically, Nightborne is one of the more involved allied races to unlock. (The Zandalar and Kul Tiran have a similar amount of questing, as you need the achievements for each leveling zone and a large chunk of the war campaign.)
Another thing to be aware of is that you will need to unlock world quests in Legion as well, as one of the Insurrection quests requires you to do some. So, if you haven’t unlocked world quests on a character on your account, you’ll need to do enough questing in each zone to get to friendly with the factions. Make sure you do that before the rep buff ends at the end of the month. (With the buff, you shouldn’t have to do much questing in each zone for the rep.)