I have already know that we can make dua in Arabic during our prayers, but can we utter the words or just keep it as our thought while we prostrate?
All praise to Allah (Swt) and blessings on Prophet Muhammad (peace be upon him, His family and Companions)
Abu Qatadah narrated that: The Prophet (peace be upon him) said to Abu Bakr: "I passed by you while you were reciting and your voice was low." He said: "I let He who, I was consulting hear." He said: "Raise your voice." Then he said to Umar: "I passed by while you were reciting and your voice was loud." So he said: "I repel drowsiness and keep Ash-Shaitan away." So he said: "Lower your voice."(Tirmidhi)
It is better to pray in a manner when a person could hear his own voice but not loud enough that everyone could hear, In this way mind is also focused and thoughts would not reside in mind In Sha Allah.
Allah (Swt) and His Messenger knows best.
"but can we utter the words or just keep it as our thought while we prostrate?"
Briefly speaking, generally vocalizing the dua will increasing the probability of granting the wishes and duas. Although it would be benefit as well if you utter your dua (and wishes) in your mind and thought, but as a rule as it is said in the hadiths that reciting dua ... , therefore the response of your inquiry could be yeah. Otherwise it would be said that ask your wishes or dua in your mind, but as much as I know and probably you know, seemingly always it is reciting the dua (or Qira'ah...) ...
As a related item:
«وَ قالَ رَبُّكُمُ ادْعُونِى أَسْتَجِبْ لَكُمْ إِنَّ الَّذِينَ يسْتَكْبِرُونَ عَنْ عِبادَتِى سَيدْخُلُونَ جَهَنَّمَ داخِرِينَ»؛
Or for instance, Imam Sadiq (a.s.) said regarding Imam Ali (a.s.):
"امیر مؤمنان (ع) "دعاه"
Amir-al-Momenan (Imam Ali) was so teller or speaker of dua
So, the following hadith shows that Imam Ali (a.s.) a person who used to say prayer (Dua). Or as you can see, in the translation of the dua (as its verb) it is written: saying prayer, hence “saying usually can be considered as talking by the tongue, not in the mind …) / although dua in the mind or thought can be profitable as well. But seemingly saying by the tongue is more effective.
During Prostration (Sujood) in Namaz, I mean Salat in Jama'ah, raising voice isn't good.
People who are beside you might be disturbed but you also should utter the tasbih in silence.
The hadith that @servant mentioned might be applicable when you are offering sunnah or nafl namaz.
But ulamaye keram of Ahlus sunnah wal Jama'ah agreed upon the fact that one should at least, utter what he is reciting yet not raising voice at all which will surely harm others' prayer in a mosque.
And Allah Ta'ala knows best.