Half of the Moon you cannot see because it faces away from Earth. Half of the side you could see is in darkness at "half Moon". That makes 3/4 of the Moon invisible and 1/4 visible. Hence "a quarter Moon" is visible.
Another viewpoint:
No, that's not really the answer. This is the answer:
Let's start with "New Moon". Later, at "Full Moon", the Moon is halfway through its cycle of phases.
Halfway between these two phases we see a half Moon. That's called the "first quarter" phase, because the Moon is a quarter of the way through its phases.
So, that's why we see the half Moon at a "quarter" phase.
It does cause confusion.
It sounds like the moon is currently waxing gibbous, with more than a quarter but less than half of the moon visible. As it progresses towards the third quarter phase, the visible portion will continue to decrease until only half of the moon is visible.
The last quarter of the Moon occurs when half of the side we see from Earth is illuminated. This phase happens after the waning gibbous phase and before the new moon phase in the lunar cycle.
Half of the Moon you cannot see because it faces away from Earth. Half of the side you could see is in darkness at "half Moon". That makes 3/4 of the Moon invisible and 1/4 visible. Hence "a quarter Moon" is visible. Another viewpoint: No, that's not really the answer. This is the answer: Let's start with "New Moon". Later, at "Full Moon", the Moon is halfway through its cycle of phases. Halfway between these two phases we see a half Moon. That's called the "first quarter" phase, because the Moon is a quarter of the way through its phases. So, that's why we see the half Moon at a "quarter" phase. It does cause confusion.
The first quarter on the moon occurs when the moon is half illuminated, transitioning from a new moon to a full moon. This phase is also known as the waxing crescent phase, where the right half of the moon is visible from Earth.
That phase is called the third quarter or it could be called a waning half moon.
first or last quarter
Your question is not clear, but if your talking about the phases of the Moon, then the answer would be a 1st and 3rd quarter.
The final half of the moon cycle is called the waning phase, while the first half is called the waxing phase. During the waxing phase, the illuminated portion of the moon visible from Earth is increasing, whereas during the waning phase, it is decreasing.
In the first quarter moon, the round portion of the visible moon faces west (forming the letter D when viewed in the northern hemisphere). In the last quarter phase the round portion faces west.
A week after the new Moon, when half of the Moon's face is illuminated, it is called the First Quarter phase. During this phase, the Moon appears as a semi-circle, with the right half lit up in the Northern Hemisphere. This occurs as the Moon moves in its orbit and more of its surface becomes visible from Earth.
These are different phases of the moon based on its appearance as observed from Earth. Crescent moon is when less than half of the moon is visible, first quarter is when half of the moon is visible, waning gibbous is when more than half but less than full moon is visible, and waxing gibbous is the phase when more than half but less than full moon is visible.
We see a Half Moon illuminated. The name of this phase is called Last Quarter. First Quarter is 1 week before a Full Moon.