Squire Trelawney Sr. is the owner of the firm Squire Trelawney & Son: Master Ship Builders in Muppet Treasure Island. Jim Hawkins and his friends seek the squire's assistance in acquiring a ship. The butler informs them that the squire is in Long Neddry for the grouse season, and not due to return until the Feast of St. Lulu. Thus, Jim is compelled to turn to the squire's halfwit son, Young Squire Trelawney (and his consultant, Mr. Bimbo).
A portrait hangs on the wall of Squire Trelawney's office in the storybook Muppet Treasure Island: Treasure Hunt. The subject of the painting looks a lot like Fozzie's Squire Trelawney character but with longer hair and distinguished facial hair, presumably a relative and possibly his father.