Nvidia nvs 440 on ebay $30 to $90 is fanless make sure you get one with cables. Nvs 450 is also fanless. The new nvs 510 has a fan.
How much heat they generate is based on dissipated power. 100 watts is 100 watts whether a card has a fan or not. The fans just get it off the card.One last thing: is it true that fanless cards heat up more than ones with fans? (read that somewhere but have no idea if that's correct). If true, does that mean more risk of overheating in the tower?
Have you considered an aftermarket GPU liquid coolant system?
4 Monitors is broad. 4 Monitors...at what resolution?
Also, if you're willing to spend what seems like a decent amount of money...a new motherboard isn't that expensive.