When Time & Materials or Cost Plus jobs also require AIA Billings, it is a little more time consuming (one extra step), but certainly accomplished.
You will still need to create an "Estimate" in QuickBooks®, you would probably prefer to call it your Budget.
When you are ready to do your billing, you would first create an Invoice by selecting the job, choosing NOT to use the existing Estimate, then click on the Time & Costs icon at the top of the Invoice window and select all of the entries from the Item, Expenses, and Time tabs. Print, save, and then DELETE the invoice.
Go to the Estimate, create the Progress Invoice, and using the invoice with all the time and costs that you just printed as a guideline, bill appropriately.
More detailed step-by-step instructions are provided in the manual, available as from your desktop as a .pdf, after you install the program.