New Jersey Gov. Chris Christie called New York Gov. Andrew Cuomo this week to complain about a Cuomo appointee's handling of a growing controversy over traffic pattern changes on the George Washington Bridge, a person familiar with the matter said.

Mr. Christie, a Republican, complained in a private phone call to Mr. Cuomo, a Democrat, that Patrick Foye, the executive director of the Port Authority of New York and New Jersey, was pressing too hard to get to the bottom of why the number of toll lanes onto the bridge from...