If you are using Cash accounting for VAT in Quickbooks and you using the purchase ledger ( Suppliers - Bills and pay bills) then if you post your payment on account you will not be reclaiming the VAT. If you only receive one bill a year , then the tax point will be the date of the bill and you wont be able to reclaim the VAT on the payments on account.
However the company you are paying should be accounting for the VAT it receives on either the date of the bill or the date payment is received whichever is the earlier, so infact ask them for a monthly VAT Invoice for the amount of your Direct Debit and that will be correct and solve your problem and their potential one.