How many quests does each companion have?
Solution 1:
Every classes first companion has three major quests that will take you off the ship; one per act.
This is in addition to the 'quests' that are your on-ship conversations with all of your crew members. These, like the three 'major' quests, are unlocked by a combination of affection and class story progression. If you get your companion affection substantially ahead of the plot curve (say, via gifts), you can use these conversations to see a nice XP boost at the start of each chapter.