No, The Congress has to initiate an Impeachment and The Senate tries the case..
BUT
The is a clause in Article V of the Constitution called the Convention of the States that allows for the States themselves to Amend the Constitution to protect against an oppressive Federal Government
It has never been convened, but there is a push going on right now (CSG) looking for the 34 States needed to try it
Search - Article V Convention of the States for more info
So basically an Article V Convention could ratify an Amendment to the Constitution to allow different rules for Impeachment
That is really simplified, and has never been tested against the SCOTUS so in theory it is possible
