Question

How should the bolded words in the sentence be contracted? Nobody is going to believe him!

Answer 1

The answer is Nobody's because Nobodys' shows possession. Nobody's is the correct format needed to show Nobody is. Hope this helps!

Answer 2

The correct option would be Nobody’s

In this case, the IS contraction is used and so the meaning of the sentences does not change. it is still expressing a future reference. 
it is important not to get confused with the possessives (´s) otherwise this sentences would not have a clear meaning e.g Nobadys´.

Nobodys´s and nobodys are not grammatically correct options. so they should be discarded at once.