Not legally, no. Any independence procedure, whether it be through a referendum or simply a grant of independence would have to be agreed on by Parliament.
In the case of Scotland, the competence to hold an independence referendum was granted to the Scottish Parliament by the government through a Section 30 order - referring to the section of the Scotland Act 1998 which allows Westminster to grant powers to the Scottish Parliament either temporarily or permanently.
With respect to Northern Ireland, schedule 1 of the Northern Ireland Act 1998 allows the Secretary of State to hold a referendum on the formation of a united Ireland should it appear likely to him that such a referendum would be successful.
It's clearly within the competence of Parliament, then, to create such a process for an individual county. Under the current law, however, any referendum held by such a locality would be unlawful, and any declaration of independence would be unilateral.