Is the english translation spoken in this way? I would translate it with: Peter hears here often cooks. Is this also o.k.?
Nope. I don't know the exact grammar for it, and from my German experience that sentence would be pretty natural in German, but in English it sounds very unnatural. "Péter hears cooks here often" or "Péter often hears cooks here" would be the two ways I might phrase it.
Stick with 'cooks'. Strictly speaking, a 'chef' is the head cook or boss cook in a restaurant, etc. Hungarian has its own word for 'chef': séf.