public boolean inCircle(int mouseX, int mouseY) {
    if (distanceFromCenterOfCircle(mouseX, mouseY) <= CIRCLE_RADIUS) {
        return true;
    }

    return false;
}

Can be

public boolean inCircle(int mouseX, int mouseY) {
    return distanceFromCenterOfCircle(mouseX, mouseY) <= CIRCLE_RADIUS;
}

(int)(Math.random() * (getWidth() - CIRCLE_RADIUS * 2));

g.setColor(new Color((int)(Math.random() * 256), (int)(Math.random() * 256), (int)(Math.random() * 256)));

Looks like you could benefit from a function that takes an integer as maximum value... and then returns an integer between 0 and max minus 1.

Which does exist - you'd simply create a new instance of Random and then call randomInstance.nextInt(maxValue).

public boolean inCircle(int mouseX, int mouseY) {
    if (distanceFromCenterOfCircle(mouseX, mouseY) <= CIRCLE_RADIUS) {
        return true;
    }

    return false;
}

Can be

public boolean inCircle(int mouseX, int mouseY) {
    return distanceFromCenterOfCircle(mouseX, mouseY) <= CIRCLE_RADIUS;
}

(int)(Math.random() * (getWidth() - CIRCLE_RADIUS * 2));

g.setColor(new Color((int)(Math.random() * 256), (int)(Math.random() * 256), (int)(Math.random() * 256)));

Looks like you could benefit from a function that takes an integer as maximum value... and then returns an integer between 0 and max minus 1.

Which does exist - you'd simply create a new instance of Random and then call randomInstance.nextInt(maxValue).

Lastly, I think you would be better off by renaming the currentCircle variable to amountOfCirclesDisplayed because that's what it really is. currentCircle seems to point to the idea that somewhere, you have a list of circles, and right now, you're working on 1 specific circle. But that's not true, because you HAVE no list of circles and all that really matters is how many you've displayed so far.