I agree with Wayne and Gary regarding what we support bringing into Sage 100. However, I think your question is more focused on how to get data out of QB rather than how to get it into MAS. This a set of free ODBC drivers you can install onto a machine running QB (at least there is in QBE). With these, it is possible to pull master file and transaction data into and Excel spreadsheet for further manipulation. I don't know if this will pull the data you are looking for specifically on payment history but it is where I would look to if you can't find it within QB.
One caveat, I am currently working with a client moving from QBE to Sage 500 (no comments about sunsets!). The ODBC drivers worked well pulling data for Customers, Vendors, Items, and GL detail history. However, we have not been able to pull detail transaction history for sales due to the volume of records within QB (I highly suspect some data corruption errors as well). We set up the extraction and let it run overnight several times without any success. We even let it run for 60 hours over a weekend once and it never returned a single record. All this to say, I can't promise success on pulling transaction history. Particularly if you are working with a really large data set or one that has a history of frequent failures.
Good luck.