I understand that one criteria for triggering companion quests is to get them to Like the player to a certain level. What other criteria are required to trigger their companion quest?
Okay, so first off, it ought to be known that the vast majority of companions do not have quests in the traditional sense. Most are recruited through a quest, but those quests don't have any special conditions for the most part. You can get each of their companion perks purely by raising their affinity for you to the maximum.
Those that do have a quest, are Nick Valentine, MacCready, Curie, and Cait.
MacCready, Curie, and Cait's requirements are strictly affinity based. All you need to do is raise your relationship status with these three, and converse with them regularly. They'll let you know when they're ready for the next step and tell you what you need to do.
Nick Valentine is the outlier. In order to do his quest - Long Time Coming, You'll need to have advanced the main quest past the quest Dangerous Minds. Specifically, after you've tracked down Kellogg, you'll need to visit The Memory Den in Goodneighbor with Nick. Additionally, you may need to first complete one or both of the other detective case files - The Gilded Grasshopper, and The Disappearing Act. You can pick up both of these quests by examining the files left lying around in Nick's office in Diamond City. He doesn't need to accompany you while you do them, but they offer some good opportunities to make Nick like you more along the way.
After you've completed those three quests, and maxed out your affinity with Nick, he'll offer his quest.
As mentioned by others, Nick's quest requires the main quest Dangerous Minds to be completed, since the events of that quest are part of Nick's motivation for the events in A Long Time Coming.
I can confirm that none of the Detective Case Files quests are required however, as I have completed Nick's quest without having started any of them (only have the 'check detective case files' miscellaneous quest entry).
All of the other companion quests only require sufficient affection to initiate.