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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Let’s look at the condition 1). It tells us that m = n = 2, because the only even prime number is 2
m^2 + 3n^2 = 2^2 + 3·2^2 = 16.
The answer is unique, and the condition is sufficient according to the Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that m = n = 2 because the smallest positive even integer is 2.
m^2 + 3n^2 = 2^2 + 3·2^2 = 16.
The answer is unique, and the condition is sufficient according to the Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).