What's the difference between the two phrases? I'm interested in nuances.
- We had no problem/ trouble finding the stolen item/ finding her a house etc
- You had no problem/trouble taking advantage of her naivety.
"To have no trouble doing something" means to find it easy or convenient.
"To have no problem doing something" means not to object to it.
In perhaps less common cases, the second one can also be used in place of the first.
Although the two words overlap, problem implies dealing with complications and difficulty.
Trouble implies more effort.
I would prefer problem in situations involving bureaucracy, especially in dealing with officialdom.
Trouble is more a case of fixing a puncture, getting the desk through the front door.
However it is a case of nuance, as you say, with plenty of room for personal preference and disagreement.
Adding on to Ronald Sole's answer to address your example sentences:
Problem and trouble both imply some kind of difficult or obstacle to be overcome. In many cases, these are interchangeable:
I had no problem/trouble passing the course
Your second example sentence, however, implies something different, that the person didn't need to overcome any moral objection. While either trouble or problem works here, you could consider a word that embodies this idea of doubt, like "qualm", "second thought", or "misgiving"
You certainly suffered no second thoughts before taking advantage of her naivety.
Naturally this depends on whether the speaker intends reproach or approval ... although this might be more a question of tone and context than verbiage.