Does Fusion Accounts enable businesses to create purely service invoices?

Business Need - I am looking for a system where the invoice does not require me to enter quantity and unit price

Within the 'Business Details' section of Fusion Accounts a business can choose not to 'Show quantity and unit price on invoice'.  If this is not required the business simply unselects this option and then clicks on the 'save' button to update the changes. The business details page can be found under the 'Business' menu.

If you choose not to show quantity and unit price on invoices, at the time of creating the invoice having chosen a sales account the business will only be asked to enter in a price for the invoice. 

So yes, Fusion Accounts does support service invoices.