This is certainly possible. Create a new shapefile by choosing New under the menu Layers and there you select Shapefile. In the options dialog you choose Line and WGS84 projection and press Ok. It will prompt you to save somwhere.
Now the newly created shapefile shows up in your layer list. Select it and toggle editing to on (the Pen icon on the toolbar, or Toggle editing under the Layer menu.)
Use the create new feature tool to draw your line. When you're done with a segment, right click to finish it. When you're done editing, press the pen icon again to toggle editing off. It will prompt you to save your edits.
Now right click on the edited layer in your layer list and select Save as. Here you can choose to save as GPX format.
Edit: As the linked question suggests, it also seems like you have to tick the "Skip attribute creation" box when saving the GPX file.