Is it correct to say open shift and close shift or is it more preferred to use begin shift and close shift? By opening shift, I mean it from the perspective of the worker that uses a cash register or cash drawer: https://www.thebalancesmb.com/balancing-a-cash-register-2890536
No, that's not normal usage.
We would say:
I am going to work the opening shift.
I am going to work the closing shift.
Or (in context), we would say:
I am going to open this morning.
I am going to close tonight.
But neither open shift nor close shift are used.