The Data Interoperability Tools .tbx appears in my system toolboxes, however when I double click quick export or import, a red X appears. I have gone into the administrator settings and the tool doesn't appear to be installed. How do I install it? I am using ArcGIS 10.2 for desktop trial. After I attempt to evaluate extensions, I receive a message which states all requested features have already been authorized on this machine.
You are supposed to install Data Interoperability extension separately which should come with your media. The trial version is available here. The Data Interoperability is a separate installation from the core desktop media. You can find it within your installation media folder, there should be a folder called "DataInterop".
After that please check in the Customize menu > Extensions - the Data Interoperability. The extension must be enabled in order you to be able to start GP tools from the ArcToolbox which are licensed separately via an extension (same thing as to any other extension such as Spatial Analyst or Network Analyst).
You have not described the process you used to install and license the Data Interoperability extension.
If you have not already done so, I recommend that you follow the Online Help for Installing the Data Interoperability extension.
If you have followed these steps once unsuccessfully, then I recommend taking at least a second run through them in case something was missed.
If that does not work then I recommend contacting Esri Support who should be able to check that the license is valid and that the documentation you are following is accurate.