Middletown township is right next to Fallsington Township in lower Bucks county, PA. Please don't post records under Chester county!
The Moon-Williamson log cabin that was built by the original Moon settler, James Moon, is still preserved and owned by Historic Fallsington, just up the road from the Fallsington Friends Meeting. They were most assuredly Quaker. whether you'll find record of the ship they arrived on I don't know.