Qu'il préfère boire? vs Qu'est-ce qu'il préfère boire?

Joseph, 

no, you cannot say ‘ qu’il préfère boire ‘.

Have a look at the lesson below on forming inverted questions starting with ‘ que ‘. 

There is a note in the lesson you linked on this : 

“ ATTENTION:

Que can be used at the start, but the statement will then be inverted (it's a more formal way of asking questions). “

 Que + [inverted statement] ? = What ... ? (French Questions)  “

And look also at the link below on how this is done specifically with il, elle, on forms.

Inverted questions in the present tense (Le Présent) in French - il/elle/on forms