Dexter's real reason for quitting his caddying job is that he's deeply affected by his first encounter with the young Judy, and he has to do something about these strong feelings right away:
But he had received a strong emotional shock, and his perturbation required a violent and immediate outlet.
Judy's overwhelming beauty and charm, along with her brutal and bratty behavior, have captivated Dexter. He can't properly express his feelings of shock and bewilderment, so he channels those feelings through the only outlet available to him: his power to quit his job.
Yes, it's a strange thing to do, for several reasons. Dexter is excellent at his job, it pays him better than other available jobs would, and he is respected and valued there. To a casual observer, there is no good reason for Dexter to quit and every reason for him to continue caddying.
His boss, Mr. Jones, even asserts that Dexter is an intelligent, honest, and grateful boy, with a good work ethic. It's out of character, then, for Dexter to not only quit his job with no notice but also to lie about why he is quitting. He tells Mr. Jones that he's become too old to be a caddy, but that's an excuse.
What this episode shows us is that Judy holds an unnatural power over Dexter. Without even trying, she's caused an upheaval in his young life: it's a pattern that will continue through the rest of the story.