If you write North Yorkshire County Council, you should write North Yorkshire County Court(s).
Edit; perhaps that needs some clarification. If the word 'court' is part of a title, i.e. X Crown Court, it is capitalised, just as you would capitalise 'council' above. If however, you were talking about 'the crown court system', or in other uses, i.e. "i'll see you in court", "The courthouse was bright red", or "you are in contempt of court", it is not capitalised.
To sum up;
Generic- 'court'
Specified (i.e. title)- Court