Trustworthiness is the quality of a person or a thing that inspires reliability. If you want a pet known for its trustworthiness, choose a loyal and obedient dog.
People or things can be described with the adjective trustworthy, and when you add the suffix -ness to it, you make the word into a noun describing the quality or trait of being trustworthy. A person can be known for trustworthiness in keeping promises, and a newspaper can be worthy of trust for correct reporting.