The standard way to implement a reversible XOR gate is by means of a controlled-NOT gate or CNOT; this is the "standard quantum XOR operation". Physics.Stackexchange
Is there a "standard quantum XNOR operation"?
The XNOR gate (sometimes ENOR, EXNOR or NXOR and pronounced as Exclusive NOR) is a digital logic gate whose function is the logical complement of the exclusive OR (XOR) gate. Wikipedia
Alternatively, what is the logical complement of the CNOT gate?
There is no "standard" method to implement XNOR, but it can be logically obtained by attaching a NOT gate (often called an X gate in quantum computing) to a logical XOR (which you know is implemented using CNOT). The X gate is applied to the target qubit of the CNOT.
To answer your question more directly, there is no standard "quantum gate" that is equivalent to XNOR. The best way to implement XNOR in a quantum circuit is with a CNOT and an X on the second qubit.
The reason why {CNOT,X} can give you a logical XNOR was explained in this answer to your own question 3.5 months ago.
