It's a subject-verb inversion needed to avoid any emphatic meaning. If you do not use such inversion it sounds very much emphatically:
"quali espressioni Mario e Giovanni usano?"
would mean: we know what expressions they use, but are those really e.g. vulgar, or mischievous, or damaging, ...? Come on!
The second version sounds quite unnatural. It's understandable, but verges on the ungrammatical. Even if you want to stress that it's Mario and Giovanni that are the focus of your sentence, you'd say something like:
Mario e Giovanni quali espressioni usano?
which isn't the finest of constructions, but is very acceptable colloquially to stress that you are ...