I will be moving to Washington state in the next couple months. The state does not have Income Tax, and I will be working from my home office. However the company I will be working for is in Kentucky, which does have a state income tax.

Will I be required to file and/or pay Kentucky state income tax?

If you are not live in Kentucky, then your income earned while in Washington state is not subject to Kentucky income tax.
Read: Your U.S. Tax Return: Working or Living in Two or More states