I guess the first question are you sure it’s the GPU fan and not the PSU or anything else spinning up to support the additional work the GPU is doing? If so and it is the GPU fan, there’s likely nothing simple you can do. The comment to reducing the GPU power by tweaking it’s power settings is probably the only easy option. But it likely will also reduce your gaming performance which might not be desirable if you’re trying to do 4K gaming.

While addressing your case airflow and other components is definitely recommended, it seems likely it’s more related to the GPU itself. Better airflow will help but it sounds like the GPU is pushing the limit of the included cooler so even better airflow probably won’t be enough to keep it from cranking the fan up. I feel like this is the standard complaint on blower style coolers. The fact they exhaust the heat out the back is helpful at reducing the heat in the case meaning you can get by with less case airflow, they also tend to be noisy especially when the GPU is dumping a lot of heat out. A better GPU cooler is likely the only real solution. But 3rd party ones aren’t cheap and of course you could mess up and kill your GPU if you’re not careful. A GPU that comes with a better cooler would be best. But again, a “better” GPU cooler is also likely going to dump more heat into the case which brings us back to your less then ideal case airflow setup.