The short answer is yes. Transactions made through PoolDues get sent to PayPal.
With QBO (online) integration to your PayPal account, those transactions will then get posted two places. The purchase information that gets sent back to PoolDues kicks off the code that is necessary based on what was purchased. For instance, if guest passes were purchased it will increment that the appropriate number, or if a membership was purchased the system will implement the code required to make the purchaser a current member. The information that gets sent to QBO will depend on how you set up the PayPal/QBO integration, but usually includes a price and product description at minimum.
If you are using the desktop version of QB, it just becomes a little more manual. Your treasurer will be able to pull reports at their leisure and import the information into QB as well.
