Does Nicola find Major Petkoff's old coat in the blue closet?
I think the answer is probably "no." Nicola only pretended that he'd found it there in order to please Catherine.
The question refers to one of the farcical moments in the play. Major Petkoff and Sergius don't know that Raina and Catherine gave Major Petkoff's coat to an enemy soldier, and the women want to keep them ignorant of that fact.
The soldier has come to return the coat to Catherine. It's in a carpet bag. But before Catherine has the opportunity to sneak the coat back into the closet -- so her husband will never be the wiser -- she is interrupted.
Where was the coat when Nicola was sent to fetch it from the closet?
Before the crucial scene in question, the last known location of the coat was inside the carpet bag. At the end of Act II, Nicola is ordered to take this bag away -- presumably to deposit it, unopened, in a foyer or cloak room. Neither Catherine nor Bluntschli has told Nicola what's inside the bag, and as far as we know, nobody else does. Everyone -- Catherine, Bluntschli, Petkoff, Sergius, and Raina -- heads to the library together. Thus, it's not clear that Catherine has the opportunity, at any point, to speak privately with either of her servants.
So it seems very doubtful that the coat had been in the blue closet when Nicola was ordered to get it in Act III. Nicola, or his fellow servant, Louka, would have had to have transferred it there earlier, and as far as we can tell, they had no reason to do so.
What seems more likely is that Nicola -- who is a quick thinker and good at intuiting what Catherine wants -- figured out that the coat was in the carpet bag. Louka, who knew about the secret visit of the soldier, may have shared this background knowledge and helped him reach the conclusion that Bluntschli had come to give Catherine back the coat. So when Catherine ordered Nicola to retrieve the coat from the closet, he took the hint. He fetched the coat from the bag, but reported to the company that he'd found it in the blue closet. This was the story Catherine wanted him to tell, and Nicola clearly decided that it was in his interest to take her side.