In the BIOS there is only fan monitoring. The CPU one has no data available. No other settings are available.
From the HP website I was finally able to update the BIOS and the Chipset drivers. Now the most recent (2023) ones are installed. Before, proceeding from "Device Manager" in Windows, it told me that the ones installed were the most recent, instead they were from 2019.
After the update the situation is unchanged, so it seems that the ordinary setting is with the fan off and that the Intel Pentium J5005 CPU type, with a very low TDP of 10W, is designed to dissipate little heat and can also work with passive cooling systems (heat sink only).
This site describes the configuration without a fan, link:
Power and thermal performance of fanless Pentium J5005 PC
How much power a 10W TDP CPU can draw?
Now I mounted the new fan on the CPU heat sink, keeping it detached from the fan header on the motherboard and connecting it to an external power supply that turns on when the devices are connected to the mains. This way it is always running without the need for temperature-based control.
I kept the old fan connected to the motherboard fan header to prevent the PC from being stopped during POST due to failure to detect the device.