what do you mean by "full instruction set"? Intel(R) Quark(TM) is a "full" x86 instruction set (Pentium instruction set). Which means it doesn't come with SSE, ...
Often binaries nowadays assume at least SSE2 which is obviously not available on Quark.
That said there is another thing preventing you from just taking binary packages (even on Edison you might run into issues).
Binaries would typically assume glibc. Whereas the default Edison Yocto comes with eglibc which is close to glibc (and hence binaries might just work) you might have a uclibc based image on your Galileo which is not so close to glibc - unless you use the Intel IoT developper kit SD card image for Galileo which is as well eglibc based.
Anyways, you most likely will have to compile for Galileo - either on the target, or in a chroot (e.g. chrooting in the target SD card) or in a Yocto build environment. With that it should be possible in principle to compile Mongo for Galileo. However, I am not so sure on the requirements - I think Mongo is quite huge, isn't it?
Not sure about CouchDB or other alternatives