Yes, as long as you're concerned about the database format they are compatible. Moreover, such compatibility is required even for "non-native" solutions like the JGit or libgit2 libraries.
I can only see two possible problems:
From time to time Git might change behaviour in a way not compatible with some of its past versions (but very rarely and with bold warnings in release notes documents long before the change is made).
So find out what versions you're running (git --version) in both worlds and if there's a major difference between them (in the X number of 1.X.Y version) consider reading the changelog for the one which has a greater X for possible gotchas.
Potential filesystem issues: these days you can mount an NTFS volume R/W when running a recent Linux kernel, and same goes about Ext2 (but not Ext3 and certainly not Ext4) on Windows but you might, in theory, accidently hit some problem with these drivers — of course, they haven't received that much love their native variants have.