No, he took the name of Pope Adrian IV.

The newly elected pope can choose to use his birth name or he may choose to use a regnal name.

Neither his birth name nor his regnal name is Peter so the answer to your question is 'No.'

The pope, upon election, chooses his regnal name. He is not required to do so and may keep his birth name if he so chooses.

when a King takes another name than his first name (this happens more with Popes) the new name is called a regnal name