This is the invoice screen in QuickBooks Enterprise Edition 6.0.  Clicking the 'Journal' icon (see my 'look here' arrow) brings up the next screen,

a print preview of the 'journal' report.

The column with the the 'look here' arrow contains the data I need to modify in each invoice (this is just one of many invoices).  What table does it come from and what field is it?  I need an absolute, exact, positive answer - I have not found any matches to 'Testing Fees' in the Invoice or InvoiceLine tables, nor have I found any codes matching codes in the Account table to connect the 'Testing Fees' account with records in this and the other 1000's of invoices I need to modify.  Help! (thanks)

I can't show you how you set up your accounting system, but for example, when using the Rock Castle Construction QuickBooks 2006 sample company file: sample_product-based business.qbw, if you look at Invoice No. 142:

You will see that the Transaction Journals for Invoice No. 142 are posted to these accounts:

To determine what Items have been used in Invoice No. 142, we run the following query:

SELECT InvoiceLineGroupItemGroupRefListID,
InvoiceLineGroupItemGroupRefFullName,
InvoiceLineItemRefListID, InvoiceLineItemRefFullName,
InvoiceLineDesc FROM InvoiceLine where RefNumber='142'

Using the ListID it's easy to see where the accounts come from. For the stock parts its:

SELECT Name, AssetAccountRefListID, AssetAccountRefFullName
from ItemInventory where ListID='90000-934380927'
or ListID='A0000-934380927'

while for the Service item (which is the type of item you are using, it's:

SELECT Name, SalesOrPurchaseAccountRefListID,
SalesOrPurchaseAccountRefFullName
from ItemService where ListID='30000-934380927'

If you look at the Chart of Accounts, you will find the Installation subaccount:

If I edit the subaccount name from "Installation" to "Contract Installation":

Then my 1000 invoices will now say "Contract Installation" too!

If you're actually asking how to change the Account used by the Service Item, edit the Service Item in QuickBooks.