Konnichiwa GaijinPunch-san!
You are right. Usually Japanese never say "どの国の言語を話しますか?" in conversation because this sentence is so formal style. I think it is on text book only.
If you want to say "what languages do you speak" formally, "どちらの国の言葉を話されます?" is more natural in Japanese conversation. But this word "どちらの国の言葉を話されます?" is familiar language and is not so formally. I think "どの国の言語を話しますか?" is correct Japanese in formal style, but "どちらの国の言葉を話されます?" is more natural.
"どの言語を話しますか" is correct Japanese, but too formal and standoffish. Specially "言語". Japanese people don't use "言語" in daily conversation. "どの国の言葉を話しますか" is more natural.
I think "何語を話しますか" is the most simple words, and the best words in daily conversation.
NANGI