No no. They will NOT ripen off the plant like tomatoes. Anyway, tomatoes are much better left on the vines to ripen unless you are expecting a killing frost.
Peppers will eventually ripen on the plant, and turn color to red, or yellow, or orange - whatever. They are much sweeter and nicer then.
However, sometimes they will mold or incur insect damage before they ripen. And your season may not be long enough for them to ripen on the plant. (I don't know where you are, although I assume the UK from your address.)
They are often picked green and can be eaten green as well. I wait until green peppers are about the size of the ones in supermarkets before I pick them green. I leave some on the plants to ripen.
I'm hedging my bets this way: picking some green and leaving some to ripen.
Pat