We are a general construction company that uses AIA billing. How do you account for an initial draw i.e deposit that relates to the total contract value?

A Cash Basis entity has Income when they receive funds; an Accrual Basis entity would track this as liability. In either case, any of the QB programs allow you to track it as "prepayment" by using "Items/Services" that allow the item to flow to your accounting, to meet your accounting requirements. The use of the Other Charge type item linked either to liability or income, allows you to operationally process this as prepayment and then to apply it to later sales, without impacting your accounting and tax reporting requirements.