Are QR codes Turing-complete under the rules of Conway's Game of Life?
QR codes are limited to a maximum size 172x172, which is not nearly large enough to built a Turing complete state machine in.
A current implementation of a Turing machine in GoL takes 1714 by 1647 cells. 
It is undoubtedly sub optimal, but compressing it to 172x172 seems a stretch.   

However let's assume the QR code is a little larger than 1714 square.
All of the elements in the listed Turing machine are constructable using gliders and besides much larger universal Turing machines have been constructed that are fully replicable.
Given a large enough QR code most of the key space is arbitrary.
There are some fixed elements that would have to be destroyed.
There are no known patterns that cannot be destroyed by a well timed glider salvo and certainly the fixed patterns in the QR codes have been explored.  

Given the amount of redundant parent patterns that reduce to the same child pattern it is quite likely it is possible.
Furthermore the error correction only has a local effect, so changing a pattern in the top corner does not affect data at the bottom.    

If you change the question to:  


  If I stamp my GoL grid with many QR codes, can I built a turing machine?  


Then the answer is yes, because its trivial to collapse a QR code to a glider and a Turing machine can be constructed using gliders and only gliders.