Is the problem of converting ambiguous grammar into unambiguous grammar computable? (Consider Domain as all context free languages).
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.
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.