In most cases, no. In older, unpatched systems, maybe, or using very specific exploits in very specific formats.
With most modern OSes, such exploits are proactively patched, and most systems have fairly sensible defaults for permissions.
Considering the relatively small number of linux systems of a specific distro, DE and version - its pretty unlikely.
In this specific case you'd need to open an image that's not an image, which would run some script or payload that would then exploit your system. Its unlikely.
If you saw a log in screen, and well, logged in, change that password - I suppose images might be handy to phish, a simple image should be harmless on its own
So, no, you don't need to reinstall your OS if you don't want to, and from the information you've given this seems excessively paranoid.