My suggestion is to start planning exactly what you want to flow between Shopify and Quickbooks (i.e. orders, product data, inventory updates, shipping/tracking info, etc.) using this free integration planning tool: www.ebridgeconnections.com/blueprint-builder.aspx
Once you have selected what you'd ideally wish to integrate, you can Request a Quote to learn more.
We have few clients that have the similar issue and we have provided them with the solution with a well reputed Microsoft solution and they are very happy. If you need help let us know.
I may be a little biased because I work for a company that uses Xero, but honestly it is the best accounting software I have tried.
I'm also a ProAdvisor for Quickbooks, and I just can't recommend them for anything.
Xero is constantly providing new updates and has an open API to allow app developers to integrate with their system. It is also really simple to use for non-accountants.