Thanks for your question. I think the others have done a good job of answering it.
It depends if the whole sentence is using "honest" correctly as an adjective. I don't have the entire sentence, so it's hard for me to say.
to show his honest opinion
to show his honest bearing
to show his honest way of doing things
The examples above are all acceptable ways to use the adjective "honest."
But, if you need the noun form with the infinitive as in these phrases:
To show respect
To show thoughtfulness
To show caution and prudence
then, you need the noun form of the adjective "honest."
"Honesty" is the noun form.
To show honesty and integrity and righteousness
Or if you want to use the adjective, you can substitute a noun clause for the word "honesty."
He has to show that he is honest.
He has to show that he's an honest person.
I hope that was my honest answer for you.