Our pet is dying.
How can i say this using names instead of our(my brother and i own this pet)
Short answer: The pet belonging to me and my brother is dying.
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My brother and I have a pet. It is dying.
The pet that belongs to my brother and me is dying.
One of the most awkward versions of this is:
My brother's and my pet is dying.
Unfortunately, from the phrasing of your question, I suspect this is the one you want to use.
CJ