I don't think English has a specific idiom for it. In the US (and I think many other English-speaking countries, but that is a guess), it is generally considered quite rude to ask about someone else about the specifics of his/her financial situation, how much money he/she earns, how much his/her car/house/other very large purchase cost.
We wouldn't really use that "delve" phrasing, but you have the right idea. The word you are looking for is "pry;" a common phrasing for general rude-inquiry matters is "
I don't mean to pry, but..." Less commonly, there is "I don't mean to stick my nose where it doesn't belong, but..." ("Nosiness" is trying to look into the dealings with others when one shouldn't. It comes from the image of a next-door neighbor standing on tiptoes with a nose over the fence to peer in at what your family is doing. It is considered an action of busybodies.)
I think "delve" is most commonly used when saying you will look into something.
"Let's delve into it later."
