Before discussing push-ups, let's start with a warning: Golfers' wrists take a lot of punishment, so standard push-ups, where the wrists are in dorsiflexion (back of hand bent closer to the forearm) could result in injuries. Be mindful of how your wrists feel before doing push-ups. With that out of the way, know that the push-up is a terrific upper-body exercise that is easy to do, and easy to know when your body has had enough. When you start to noticeably slow down when pushing upward, or your form gets sloppy (keep the spine flat/neutral, elbows tucked at a 45-degree angle with your trunk, feet close together and core muscles contracted), it's time to take a break or move on to another exercise.
For golfers, the true benefits of push-ups are that they strengthen the pectoral (chest) muscles and improve stability of soft tissue in the shoulders. The rotator cuff muscles and deltoids play a key role in controlling the positions of the club as you swing, but also in safely stopping the club. If you've ever seen a golfer who lets the width of the club collapse at the top, can't compress the ball with any real power, or look like they are swinging out of control, they could really benefit from more push-ups.
Many people find push-ups boring and that problem gets worse as they get stronger, because they have to do more and more push-ups to even maintain gains from previous workouts. Boredom is certainly understandable. But before you ditch this valuable body-weight exercise for something else, try this push-up progression from the video below. See if this doesn't reinvigorate your interest level while also challenging you in ways a standard push-up can't. Just remember the warning about wrist protection. That should be your first priority.