The FPGAs that Blackmagic uses are quite sensitive to temperature. (AJA is in the same boat.)
The quad is rated max 40C which is not that high.
For reference that 8K pro uses a chip that's slightly more tolerant up to 50C. Even when running 720@59.94 this chip heats up to the point it almost maxes out without adequate ventilation in the case.
Personally, I wouldn't dare to run a Decklink card without a fan unless I had a significantly larger heatsink.
The FPGAs do include a temperature sensor but Desktop Video doesn't display it. For that you'll need to download the Decklink SDK and compile the sample status application which includes a temperature readout. With this you could test temps with the fan off.
Alternatively you could just change to a quieter fan like you already mentioned. I put an Enzotech SLF-40mm on my Mini Recorder 4k and it's been great ever since. I think the quad HDMI uses the same spacing so the SLF-40 would work as well. (Discussion here:
viewtopic.php?f=3&t=107148)