Can we claim our solar panel system for tax credits if we did not pay anything down, but are instead making monthly payments?

If it is a lease arrangement, you cannot take the tax credit.   But if you financed the purchase of the system (you are the system owner), you can take the tax credit for cost of the system.   You can't include interest or finance charges, but the cost of the solar system does qualify for the credit.  The key is ownership.  

See Federal Tax Credit for Residential Solar Energy