Checking CLion help https://www.jetbrains.com/help/clion/2016.2/importing-existing-non-cmake-project.html I see how to import a non-CMake project into CLion.
And I'm also able to clone a project from GitHub https://www.jetbrains.com/help/clion/2016.2/cloning-a-repository-from-github.html
The project https://github.com/quickfix/quickfix uses ./bootstrap and ./configure to setup a makefile.
What I'd like to do is import that makefile into my CLion project and build and run from that. Is this possible?
While it is possible to "import a project" that's not CMake-based into your CLion project, CLion does not itself directly support using Makefiles as an origination point for a project yet. I know that this is something that has been wanted by many people, and as far as I know, the creators of CLion are at some point planning to integrate some support for this.
In the meantime, however, there is no handy way to do this directly. CMake is a build system configurator, in that it generates its own set of Makefiles to build everything, based on the things you write in your CMakeLists.txt file.
Your best bet, should you want to use the quickfix lib in a project of yours, is to learn the CMake process for building an external dependency, and then linking it to your project. A good blog post on such a thing can be found here. If you simply want to work on changes to it in CLion for your own convenience, but keep the original build files, you could just have CLion generate its own little CMakeLists.txt file for the purposes of importing and color-coding everything, and then set your debug config, etc to point to the binaries generated by running make in the command line.
Long story short, there's no easy way to do what you are talking about directly, but depending on your intended purpose, there are a couple of alternate paths to a similar end. Hope this helps!