Yes, typically console game developers will use a game engine that works on both PC and on their target console.

They can then develop the game mainly on their desktop PCs, and also run/test it on the consoles as required.

The general public doesn't get to do this, because consoles are generally "closed off", so that you can't develop your own programs on them. Typically, you need to sign a licencing deal with Sony/Microsoft/Nintendo, and buy special "development kits" from them, which are similar to the consoles that the general public buys, except they allow you to write your own software for them.

The exception to this situation is:

1) The "homebrew" scene. These people have hacked and reverse-engineered their consoles and the required software, so they can defeat the mechanisms that stop the general public from writing their own software. This is of questionable legality (you won't be arrested for it, but it would be hard to get your homebrew game published!).

2) XNA for Xbox360 -- AKA XBLIG -- AKA Creators club. Microsoft released a software development kit called XNA, which is basically a game engine that runs on Windows and the Xbox360. You don't need to be a professional to use it, and you don't have to spend a fortune on special development kits of licensing (your regular 360 can be used).