You can use the same feed, with the following caveats:
The nugets built from the two branches obviously have to have different (non-colliding) versions. I add the 'dev' suffix to the package built from the 'dev' branch (eg 1.2.3.4-dev), and leave my 'stable' branch bare.
You have to be explicit when you 'create release', because by default Octopus will pick up the highest version nuget available, and that might not be the release you want (stable presumably lags dev). Pick the version of the package you want (and set the deployment version appropriately). If you are creating your release via TeamCity, ensure you use the --packageVersion argument as well as set the release number.
Since the above only works if there's only one package in the release, your deployment process does (unfortunately) have to produce one monolithic nuget, or you will get version-mismatches.
The advantage of this kind of arrangement, of course, is that at a pinch you could push your dev build to QA (or Prod) if the need ever arose.
All this assumes both branches build as the same package of course. You could build different packages between the dev and stable branches (but I don't think I'd recommend this because of duplicating all the Octopus config).
Update: apparently you can use Octo.exe to specify different version numbers across different packages - see https://github.com/OctopusDeploy/Octopus-Tools.