This should be possible. To accomplish it follow these steps:
1. Configure the GpsGate input to use the Holux BT 236.
2. Add a Garmin emulator to the output. Select an appropriate port and in the Garmin emulator settings dialog, either enter the unit id manually (you can find it using MapSource) or start the wizard. If you are entering it manually, click ok and you're done. If not:
3. Let the wizard search the COM-ports so that it can find your GPS. Make sure bluetooth is unchecked, since it's time consuming and won't find anything.
4. When the wizard is done, you should see your built-in GPS in the list of devices. Select the built-in GPS and click ok. If your GPS isn't in the list, run the wizard again and check that it looks for the unit id on the appropriate port (it prints a message telling you what it is doing). If it still isn't found, you need to enter the unit id manually. You can find the unit id using MapSource.
5. You should now see a unit id in the Garmin emulator configuration dialog. Click ok.
Now the Garmin emulator is configured to be used with Garmin Que. Next you need to configure Que to use the GpsGate virtual port instead of the internal GPS port. You need to check in the Garmin Que manual to find this setting. I believe the port can be configured under "Settings/System/Interface" or something like that.