Distrust - like truth, hope,... - is an abstract noun.
This link from UsingEnglish may help you (or confuse you even more, because abstract nouns can be countable or uncountable, depending on the function of the noun in each sentence - id est, subject, direct object, etc...):
Abstract Nouns - Glossary Definition - UsingEnglish.com
What I can tell you is the following:
- You can never say "a weather", but you can say "the weather".
- You can use "distrust" with the articles "a" and "the" and without any article. Here are some examples:
"Which features in the constitution reflect a distrust in democracy?"
"Can I get over the distrust regarding my husband?"
"Distrust of police is, after all, highly unsurprising in Police States."
"Willingham explains his distrust of the media."