The answer is yes. Even better it may not be as complicated as you think.
We use the "quadruplet" notation when we want four notes in the space of - usually - three notes. You question implies that you have a good understanding of how that works. We need this notation because otherwise there is no simple way to notate the actual note lengths required without making the typography look very confusing (the actual note lengths in the quadruplet are three quarters of whatever note length we are talking about).
Now sometimes you need four notes played in the length of a different number of notes. So you would do this if you wanted four in the time of five say or four in the time of seven. You have no need to do it for four in the time of one or four in the time of two because standard notation covers this. For example if you want four notes in the time of two quavers (eighth notes) then you just add another beam to give you four semiquavers (sixteenth notes).
So this type of notation is normally used where standard notation would make it difficult to read.
Same for other groupings? Yes, absolutely. You can do whatever you need to do. Typically the notes are all beamed together and there is a slur over them with the number against it. You'll have to trust me on this but when you get used to it it's normally obvious what is required.
This gets used a lot, especially in certain types of music. Chopin does it frequently for example, as does Scriabin.
Hope that helps