Offhand my guess is no. Even though you're looking at 2 monitors... aren't they the same desktop view? It's kind of like asking if you can have 2 wallpapers at the same time on a single monitor. However, many (or most) Linux distros do allow multiple workspaces, usually preconfigured for 2 or 4. Each of these is a somewhat independent desktop, and you can often find in the desktop settings that it will allow different wallpaper in each workspace. So you can click between them to use different workspaces for different purposes, and the wallpaper would be an immediate clue which workspace you're on... but I've not heard of the ability to show two workspaces simultaneously on the same monitor, or across two or more monitors.
Ah well, a little Googling says I may be wrong. That won't surprise me (or my wife)... it is easy to be unaware of things we never do!
But a quick look says that it might be tricky to set up. I'll let you investigate further on your own. Maybe you'll find a simple way, or maybe you are willing to work out the details needed.
But another quick/easy solution that might work for you is to run a virtual machine. Your VM could have another instance of your same version of Linux, or something different, and you could open it full screen in its own workspace. That would give an appearance at least of two independent desktops, and each could indeed have its own wallpaper that way. I don't mess much with VM's either, and I can't always get them to open full screen, but some of them do. So this could be a possible solution.
Good luck!
