No, I think this is part of the internal engine coding, not a user setting.
Also, why would you need to do this? The game saves a nice uncompressed PNG of each screenshot, and if you use Shift + F12, it will attempt to save a 4K Screenshot, even if you don't have a 4K capable computer.... Those "Hi Res Screenshots" at 4K are almost 45mb each!
I guess there are some benefits to having each screenshot uniquely named for easier identification later on, but even with the hundreds of screenies I've taken, I haven't got them confused or not know what each was, because I worked hard on the map, there are timestamps of each screenshot, and you can easily create folders for them in the AppData folder if needed, and then you can move/copy your screenies to each folder you make or do whatever...
It would be kind of neat if the screenshot had <mapname>_screenshot as the filename template, but I can see this going bad with strangely named Cities (people who use special characters in city names)
Valve do this for all source engine games, the name of the bsp map is the name of the screenshot, with incremental numbers at the end.
However, I don't see this adding much value to the overall game or the overall "taking screenshots" experience.
TL;DR:
Not an option you can change
You can manage the screenshots in Windows explorer easily
If you care about file extensions/types, download and install a free program called SageThumbs, which adds many features to the display of thumbnails in Windows, but also adds "Convert to JPG, Convert to GIF, Convert to BMP and Convert to PNG" to your right click menu in Windows explorer, then you can convert these PNGs to other common formats, and resize them using Paint.NET (another free program, an image editor with almost as many features as Photoshop)
Hope this helps
Sagethumbs:
http://sourceforge.net/projects/sagethumbs/
Paint.NET:
http://www.dotpdn.com/downloads/pdn.html