"Can I compile a c++ program under linux to make a .exe to run on windows "

Yes :

I confirm the last post and cross compilers can natively be found with recent Fedora (and any linux distribution, messeems!) The way all the stuff necessary could be downloaded and made working can be found with adaptations of :

cran.r-project.org/doc/contrib/cross-build.pdf 

(one just has to stop at "make xtools": they are downloaded and put into a proper place).

Else, if you like linux and have "only"  Windows, you can use cygwin (avoids dualbooting or switching to another computer)  with the sources (it is very rare that sources need to be different between both systems, they use the same compiler -gcc/g+++- and almost the same include file names.... ) ...

"and viceversa"

No (not because linux executables do not have a .exe suffix!):

 but you can edit/compile/test your program(s) under cygwin (google search can find it) until it works and port it (i.E the source, the compilation  and test scripts) to Linux (there is almost no effort, apart the efforts linked with different disk nomenclatures, of course).

 Else, unless emulating linux with vmplayer/qemu/vbox -all these emulators are Windows ported-? but it is more complicated than C/C++....