The correct option is D marrying
Let’s examine the sentence: “She has no intention of ____ a poor man like you.” The sentence specifies ‘intention’ or ‘thought’ of not marrying. As the sentence talks about the thought of ‘marrying’, and not the action of ‘marrying’, and so it requires a noun or a gerund in the blank. The gerund always has the same function as a noun (although it looks like a verb). The gerund ends in ‘ing.’ As Option C fulfils this requirement, so it is the correct answer. All the other options cannot be used as they would make the sentence meaningless.