Peter was born in Valencia, to the house of Barcelona, crowned in Zaragoza (King of Aragon) and again in Palermo (King of Sicily). He corresponded with peers across Europe and the Mediterranean world, enjoyed the work of troubadours, and may even have composed a sirventes or two himself. His wife Constance was born in the Kingdom of Sicily, a veritable melting pot where Arab, Greek, Lombard, Norman, Swabian and many other languages and dialects were spoken, and raised during the days when the Scuola poetica Siciliana was still one of the dominant cultural forces in Europe.
What would their native languages have been? What would their common languages have been? Did Peter and his court speak Valencian, Aragonese, Catalan? What about Sicilian? Occitan? What language did he compose (or have ghost written) his sirventesos in? When Charles I of Anjou wrote him to suggest they settle their dispute over Sicily by personal combat, what language would it have been in?
The officers in the court of the Crown of Aragon where supposed to know aragonese, latin and catalan. I don't know how he would have replied Charles I of Anjou, but the monarchs of the Crown of Aragon used to replie to Castilian letters in castilian.
The language people spoke in both the Principality of Catalonia and the Kingdom of Valencia was catalan, which nowadays is also called "valencian" by some valencians. In the case of Peter II , it is possible he spoke in aragonese, as it is depicted by his father James I in his own chronicles, written in catalan, which used to be the spoken language of the kings of Aragon and their court and office at least until 1412.
He wrote his sirventesos in occitan, as was the fad at the time (even Dante wrote them in occitan in the Comedy).
I really can't be sure about Constance's main language but I suppose it would be the same as in the court, or the same as his father. Sicilian is my guess, but I'm not sure. I supposed both where able to understand themselves with their knowledge of catalan and italian languages, because of the similarity, and also think that high nobility used to have personal mentors to prepare them for this.
I would advise you to keep in mind that the kings of the Crown of Aragon (a historiographic concept to refer to a composite monarchy which had no name of its own, but was referred as "the states or lands of the King of Aragon") didn't get numbered until Peter III, the one who lived from 1336 to 1387, so I would advise you to not use them in a fiction, for example. Also, Peter III called himself Peter III because he was the third Peter of the catalan dinasty, but there was one Peter before the Counts of Barcelona came in that was King of Aragon, so in Aragon call him Peter IV nowadays, and Peter III would be the Peter II which you asked about.
Also, bear in mind their that those kings where called Pere, Jaume, Alfons o Anfós, etc. instead of Peter, Jaume, Alphonse, or Pedro, Jaime, Alfonso (sorry if this is obvious but a lot of times I found of people writing of "Jaime I" IN ENGLISH, where the rule would be to keep the original name or translate it to your own language).
Hope this helped a little!