I doubt they will exchange the 20 inch monitor.
I doubt they would exchange the 20 inch monitor.
Which is correct, and why?
Both are acceptable, but the usage is slightly different. The following extensions to the sentence illustrate the most common usage:
In general, would is used in situations hypothetical, or contingent on some unfulfilled condition; will is used in situations that are more certain. The first sentence takes will because the speaker definitely intends to take the monitor back. The second uses will because, although there is a condition, there is a definite possibility of it being fulfilled. The third sentence takes would because the speaker has no intention of taking it back—and therefore the condition is entirely hypothetical.
The implication of the sentence is that you want to trade in your monitor and are wondering if the store will allow it. In that case "I doubt they will exchange the monitor" is correct, using the future tense (since it is something that happens in the future.)
"I doubt they would exchange the monitor" is not correct. In this case you would need a past tense on doubt, "I doubted they would exchange the monitor."
There is a rare future subjunctive form in English that you can use if you want to express doubt (though this seems unnecessary since you express doubt explicitly.) You could say something like "I'd be surprised if they were to exchange the monitor"
Well, it is difficult to distinguish as the context is missing, I mean, what does a monitor has to do with anything mentioned in the previous sentences?
Furthermore, the last sentence is broken, or at least an article is missing there. Either they "will exchange a 20-inch monitor", or "they will exchange the 20-inch monitor".
Grammatically, both sentences are correct, but depending on the context, which we don't know here, one of them would be more suitable.