Hahaha, nowadays this is a very used question in Spain, because people start going out without being in a relationship ("friends with benefits"), and only after some months they can ask each other or decide if they are in a relationship, so I thought this is a very useful phrase!! LOL
"Are we in a relationship" was accepted. I have a feeling that if Duo accepts it and it is not a word for word translation, then it is the right one. Jojouichanco, above, tried "Are we related" and failed. I think this is a strong sign that the correct meaning is "are we dating?"
To me, "Are we related" sounds like you're asking someone if they are a relative of yours. Perhaps you are at a big family reunion and you see someone you don't know. You might ask, "Are we related?" The answer might be, "Yes, I'm Peter and Carol's daughter." Or perhaps, "No, I'm dating Peter and Carol's son, Nick."
When talking about a family member, one can say, a relative:
die Verwandte , der Verwandter = the relative
die Verwandten = the relatives
I can't really think of when anyone would ask, "Do we have a relationship?" If you're at a party, and you introduce yourself to one of the guests, a more natural question would be, "How do you know John?" There are many possible answers: "He's my brother" or "We work together" or "We went to school together" or even, "I don't know him. I came with my friend and he is in John's book club."
The way I understand it is that this phrase translates to something along the lines of "Are we going out?" or "Are we dating?". I believe they're using the word relations in a way you'd have intimate relations with someone as opposed to just being related to a person family wise... if that makes sense... I'm sleepy...
As a native speaker I would say that "Are we in a relationship?" is far too formal. You would generally use the term 'relationship' to describe other people or your own position to a third party. If you were speaking directly to your partner it would be more natural to say "Are we dating?" or "Are we going out?".
To be honest though, it's not a question you should have to ask.
The "h" in front of a vowel is generally pronounced, while behind it serves to stretch and open it. In the case of "Beziehung", I think it's not pronounced because it belongs to the vowel in front. Note, that the "ung" is a suffix and "zieh" comes from "ziehen" (to pull/draw). In words like "Freiheit" (where "heit" is the suffix) or "Seehund" you would pronounce the "h".