There was an intense professional rivalry, and it became a serious issue when the latter received priority of supplies over the former. Patton was not the most diplomatic sort of person, and made his opinions over what he felt was priority to a poor plan absolutely clear. Patton famously was ordered to go over to the defensive during the time Market Garden was set to start, but also famously interpreted those orders quite loosely, launching attacks to dress his line and prepare for further operations. The Patton Society, which works closely with the Patton Museum, has this telling quote:

On September 17th, Patton wrote an entry named "Operation Market-Garden" in it he wrote "To hell with Monty. I must get so involved that they can't stop me. I told Bradley not to call me until after dark on the 19th. He agreed."

It should be noted though that Patton also frequently attacked his commander, Bradley, even when the latter probably didn't deserve it. Patton's ego and flair for self-promotion is matched only by Montgomery's, so lend only a skeptical and objective eye if you decide to read either's journals. Montgomery survived the post-war, and you will find that his memory on certain subjects (e.g: the overall significance of Caen) did not survive with age.