No. You can't dual wield Unarmed Strikes because they are not considered a Light weapon. That said, I don't believe it would be gamebreaking to houserule it that way.

However, in any case, you can't use two-weapon fighting and the Monk's martial arts on the same turn, because each uses a bonus action, and you only get one bonus action per turn. So your example of "tri-wielding" with a quarterstaff attack, an off-hand Unarmed Strike, and then a Martial Arts bonus Unarmed Strike would not be possible, because it would require two bonus actions to pull off.

Two weapon fighting:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in your other hand.

Martial Arts:

When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.

Note that the Monk Martial Arts ability does allow the Monk to effectively dual wield Unarmed Strikes, since they can unarmed strike as their main Attack action, and as their bonus action. And unlike regular Two Weapon Fighting, they get the bonus to their damage roll on the bonus attack as well.

This is specifically stated as something you can do in the ability's description as long as you have a ki point to spend. And what's more, you can make three attacks per turn with it:

Radiant Sun Bolt

You gain a ranged spell attack that you can use with the Attack action [...] When you use the Attack action on your turn to use this special attack, you can spend 1 ki point to make two additional attacks with it as a bonus action.