Fiddler allow you to capture HTTP traffic acting as a proxy between you and network.
So to capture WMS/WFS requests from any software I believe you have just to run Fiddler as usual. The same way you have done it with MapInfo - no additional configuring is required.
PS I have tested it with ArcGIS Server - successfully capturing REST traffic.
You was right - running fiddler without additional configuration of QGIS did nothing.
I was able to capture traffic from QGIS by navigating to QGIS Settings\Options\Network and checking
Use proxy for web access. Then I have setup
127.0.0.1 for Host,
8888 for Port and
HttpProxy for Proxy type.
You can use these parameters too or check what Fiddler had setup in Windows Control Panel\Internet Options\Connections\LAN settings\Advanced
I think this is happening because QGIS is not reading system Internet Properties and MapInfo is reading.