Let me try and summarize my understanding of the comments above. There is a Turing machine that given a presentation of a group will list all the presentations of the group. Thus, if a problem is decidable or positively-decidable (for instance the word-problem) for one presentation it is decidable or positively-decidable for all presentations.

This implies that if you have a presentation that is undecidable whether it is trivial, it must be of a non-trivial group. This means we cannot have an explicit example of such presentation.

Am I right?