If you had two viruses in play, one that infected Linux with a payload to infect Windows, then it could conceptually happen. However, a native Windows virus cannot run in Linux at all. The reason why nothing would happen has to do with something known as the "Application Binary Interface", or "ABI" for short.
In Windows, most system calls are performed through something called "INT 20h", while in Linux, system calls are performed through "INT 80h". This means that a Windows-based virus that tries to call the system through 20h would simply terminate without causing any harm, as INT 20h is a "DOS-compatible terminate program" command.
In reality, most virus writers are going to go through the path of least resistance: write a Linux virus to infect the currently running Linux system, and write a Windows virus to infect the currently running Windows system. It would take a significant amount of extra effort just to write a virus whose only purpose was to infect an offline OS.
As far as I know, it would be far easier to simply write a hypervisor virus that runs "underneath" the OS, as then the ABI would have no meaning, since they'd have direct access to the hardware and memory.
On the other hand, if you download a virus intentionally, for example, you were trying to get a copy of a game from a questionable resource, and you did this in Linux, but downloaded the file to your Windows partition, and you later ran the program in Windows, then you could indeed get infected at that point. The main point, however, is that Linux is impervious to Windows viruses, and Windows is impervious to Linux viruses. They simply do not speak the same language.