Yes.  Kentucky accepts tax payments online and by phone.

Online:  https://epayment.ky.gov/EPAY

Phone: (502) 564-4581

You can use the online method as long as you are paying by the due date.  Estimated Tax Payments cannot be made via online option.

For more information see page 2 of this pdf from Kentucky.  http://revenue.ky.gov/NR/rdonlyres/94DECC98-59AB-404B-8EE9-8D71804B3CFD/0/FAQsIII2015.pdf