Can I use some kind of adapter HDMI to VGA with the Jetson Nano?

VGA is from the days when monitors could not auto configure. Basically, you either used some default mode, or you installed a “driver disk” which was a list of the monitor’s specs, and the driver would read those specs.

After this, using digital interfaces, a “DDC” wire was added. This allowed the video card to ask the monitor what its specs were, and this was the first plug-n-play for video.

On a desktop PC you can set modes manually (such as required by VGA), or let it auto configure if a more modern connector is present (such as HDMI, DVI-D, or DisplayPort). The embedded Jetson systems do not allow manual setting, and work only with auto configuration, so it seems the default resolution does not coincide with something the monitor supports.

Mostly I suggest switching to true HDMI. You could modify your kernel source and put in a new kernel which has a default fallback mode your monitor supports, but that’s a lot of work. If you want to go that route, then see this thread:
https://forums.developer.nvidia.com/t/1440x2560-hdmi-display-not-working/77764/2