Hi Frances,
However, since you have ArcGIS on your computer, there is a much simpler way to proceed. An MXD file does not contain any data layers or other spatial information. Instead, it is just a list of addresses where the files containing this information can be found on your computer. As a result, the quickest way to get any information from an MXD file into a QGIS project would be to open the MXD file, find the data layer you are interested in, and then open its properties (by right-clicking on its name in the TABLE OF CONTENTS window). In the LAYER PROPERTIES window that opens, you can find out its file format and where it is stored on your computer. If it is stored as a shapefile or an ASCII Raster, you can simply go to the location where it is stored on your computer and load it into QGIS. If it is stored as a ArcGIS Layer File, you will need to save it as a shapefile before you can use it in ArcGIS. This can be done by closing the LAYER PROPERTIES window, right-clicking on its name again in the TABLE OF CONTENTS window and selecting DATA> EXPORT DATA. If it is sorted as a ArcGIS GRD file, you will need to convert it into a Geotiff before you can use it in QGIS. This is done using the same EXPORT DATA tool.
I hope this helps. If it doesn't work, or give you the results you are looking for, just post back on this thread and I'll see what else I can suggest.
All the best,
Colin