Is converting an ambiguous grammar to an unambiguous grammar computable?
I assume that question is about derivation, otherwise the outcome does not change, but please read Yuval's comment for further details.

No it is not computable in the domain of your choice. There are inherently ambigous grammars, which are subset of context-free grammars. For the subset without inherently ambigous ones it is computable.

I like to point to thesis about ambiguous grammars, which is consistent with what I wrote, contains the same proof as TCS topic.

Since the problem of finding out whether grammar is inherently ambiguous is undecidable then you cannot build halting Turing Machine for such task. And yes, if you could built one it would be decidable.