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In classical electromagnetic theory, If parallel plates are charged oppositely and placed close to each other, there will be no charge will not flow from one plate to another.

How does this situation change if one considers Quantum electrodynamics? Can the electric field in between the plates cause pair production? What is the probability, if it happens? How does one apply the formalism of quantum field theory to such a question ? I am rather new to the subject.

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Yes, the effect you're looking for is called Schwinger pair production. It requires immensely strong electric fields (of the order of $10^{18}$ V/m) for a constant field.

One of the methods for computing the rate is the worldline method, described briefly here. To follow it, some knowledge of effective action methods are required.

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