I recently asked myself that same question!
The most horsepower I found so far in a silent/passive gfx card is this one:
ASUS GeForce GT 1030 2GB GDDR5 (GT1030-2G-CSM)
According to NVIDIA, this chip is plenty capable of driving a 4k monitor, but the spec of the card itself is quite useless! And I found only speculations on 4k on various forums…
It’s only about $100, so I’m considering buying one to test it myself…
The DVI-D connection is limited to 2560x1600 (2k) but the HDMI output could in theory do 4k and even 8k.
Some other cards have active fans that completely halt when the workload is "light enough ". A nifty idea, but I don’t trust it to not switch back on at a bad moment! Murphey’s law…
Please keep us informed of your findings.