It's colloquial language of the workplace. "Quality work" is clear. "High quality" is often shortened to "quality." "Quality work" means good work, well done work, high quality work.
"Out the back door" or "out the door" is a colorful way of saying "shipped." In either case, it means COMPLETELY finished, ALL steps completed, the product is on its way to the customer.
"The back door" refers to shipping. The back door is where the shipping dock is.
If a company is manufacturing something, there is a point at which the product has been assembled, but is still inside the building waiting for other things to happen. Sometimes that's a long wait. Maybe it needs to be crated. Maybe the instruction manual needs to be translated. Maybe it needs a final quality inspection.
From the point of view of some people within the company, the work seems "finished," but it isn't, really. The customer doesn't pay until they get it.
When the product goes "out the back door" to be loaded onto a truck, that's a critical moment.. The customer still doesn't have it, but the responsibility has shifted. The company has completed all of its work, it is REALLY "finished," it is now the shipper's responsibility.
Figuratively, even if it's not a physical product, the phrase "out the door" is used.
In a software company, one could imagine this conversation:
Manager: "Is it finished?"
Engineer: "Yes."
Manager: "Why hasn't Megacorp received it?"
Engineer: "It's in QA [quality assurance]."
Manager: "That's not good enough. In QA doesn't count. You have to get it out the door."