While kneeling, a simple way to activate the quads is to use your legs to lift your hips off of your heels.
Quads activated to lift the hips (or to keep them lifted).
Note how belly is pulled in!
Your quads will activate. You can then slowly sink your hips down while keeping them active.
To anchor the quads from above, keep your belly pulled in while at the same time creating an upward pull on the front of your hip bones.
If you can kneel comfortably with your bum on your heels, you can activate your quads without actually causing your body to lift. A simple trick to cause your quads to activate while kneeling is to press your feet down into the floor. I'd strongly suggest slowly and smoothly activating and then slowly and smoothly relaxing.
If your knees tend to be uncomfortable while kneeling, this can be one way of helping to make them comfortable.
A variation of kneeling and stretching the quads is to try to separate your feet enough that you can let your butt sink towards the floor between your feet. In yoga terminology, this is called hero pose or virasana.
Note that this foot position, and it's counterpose, lotus, are both achievable because of the ability of the knees to rotate when bent. (It may actually be better described as rotation plus translation).
Here again what you could do is activate your quads, as if to resist your hips sinking down, and then slowly relax.
To make your knees comfortable while stretching the quads in hero, try varying the distance between your knees. Move your knees outwards or inwards and use the position that is most comfortable.
If you are comfortable with kneeling or with hero pose, another quad stretching option is to tilt your pelvis back while bending your spine forwards.
Your abs will activate and this will automatically create a pull on the pubic bone and ASICs, helping to anchor the quads from above while you stretch them. Leaning back, you may find that your toes tend to naturally press into the floor. However, you have the option of consciously pressing them down so that you anchor your quads from below the knees.
Note that an option in the hero variation of this stretch is to place a yoga block or stack of books between your feet and sit on them whether upright or doing the "Bent back" version of either quadriceps stretch.
For an even deeper quad stretch, and this assumes that you are comfortable with kneeling, hero and the bent back version of both, then a further option is to try and lay back on your back.
You can try leaning back onto your elbows first.
The trick when resting back on your elbows is to keep your abs engaged. This can be especially important the further back you go because this position can be quite intense on your lower back. And if it is the case that it is too intense, then leave it out. Or focus on creating a pull on your ASICs and/or pubic bone towards the front of your ribcage or sternum. To make this a bit easier you can try drawing your earholes away from your shoulders as you lean back so that your neck feels long also.
An alternative to stretching the quads with both legs in hero pose is to have only one leg in hero pose.
One option is to straighten the other leg. If your hip on the hero leg side doesn't touch the floor you'll probably need to use the hand on the straight-leg-side to prop your body.
As you lean back you may find your hero leg knee tends to lift. If you are on your elbows you can from there wiggle your hips forwards enough that the hero leg knee touches the floor.
So that it's easier to create an upwards pull on your pubic bone and or ASIC (on both sides or on the hero-leg-side) you can use your elbows to anchor your ribcage and from there use your abs to pull on the pubic bone and ASIC.
Another quad stretching option with one leg in hero is to bend the other knee and place the foot against the inner thigh of the hero foot in a variation of the bharadvajasana pose.
One option is to sit upright and press down with the non hero knee while at the same time resisting with the hero leg.
Optionally, you can try bending to the hero leg side. You can use your hand to support your body weight or lift your hands to add weight to the stretch.
You can also twist away from the hero leg side and then towards it.
In any of these positions you can try shifting weight to the hero leg side while at the same time resisting by pressing the toes or foot down or resisting with the quadriceps fo the hero leg itself. The trick here is activating the quad but not with enough force that it pushes your torso to the opposite side.
One non kneeling option for stretching the quads is to squat. One option is to squat with feet flat on the floor and then lift the heels. Another is to lower down into a squat with heels lifted.
If you have difficulty with squatting, say due to lack of balance, try the suggestions in deep squats
In either case, the lifted heel position will tend to force muscles of the foot and ankle to activate and that can actually help to anchor the quads from below.
To further anchor the quads you could keep the chest lifted (and/or spine long) to create an upwards pull on pubic bone and/or ASICs. Or simply pull your lower belly in using your transverse abdominis muscles.
This will naturally add tension to the obliques and rectus abdominis, thus creating an upwards pull on the pubic bone and ASICs.
While standing, the usual way to stretch the quads is to stand on one leg and grab the other foot behind the back. Standing upright, the general idea is to move the lifted-leg knee rearwards to stretch the lifted-leg quadriceps.
To anchor the quad from above you can stand with spine long, chest lifted and pull upwards on the front of the hip bones. Or as mentioned previously, with spine long and chest lifted, pull the belly in so that you activate your transverse abdominis.
Another option is to use the muscles of the standing leg to help tilt the pelvis rearwards.
Not forgetting to anchor the quads from below, you can do that by making the foot and ankle of the lifted leg active or strong.
A variation from this position is to bend forwards. This can be challenging, particularly if your hamstrings are tight. The focus may then switch to the standing leg and the discomfort of the hamstring stretch.
Another variation of the above pose is to reach the bent knee leg back. This decreases the knee bend but increase the back bend of the hip so that the focus is more on the rectus femoris portion of the quadriceps.
Here again the suggestion is to keep the foot and ankle active and as well maintain an upward pull on pubic bone and ASIC via the transverse abdominis.
Here also, you can make the quad stretch a little more challenging by trying to maintain it while bending forwards.
Another group of quad stretches that could be called semi prone involve having one leg back. The front leg can be bent in a lunge.
From there the idea is to bend the back knee, grab the foot, and if shoulder mobility allows, turn the hand so that the fingers face in the same direction as the toes.
From there the idea is to use the hand, and arm, to press the foot down.
Once you can get the hand in position, rather than just trying to press the foot down, actually resist the foot against the hand. Resist with just enough force that your foot, ankle and quadriceps are active. But not so much you actually push the hand back.
Obviously you will have to exert opposing pressure with the hand and that will require an anchored shoulder and ribcage.
A variation of this basic position is to have the front leg in pigeon.
Another variation of this quad stretching position is to have the front leg straight as if working towards the splits.
Even more challenging are fully prone quad stretches. Here the idea is the same, to use the hand to push the heel down.
You can do this one leg at a time or try to do both sides at the same time.
Because of the difficulty of this position this becomes less about stretching the quads and more about coordinating the different parts of the body involved.
A variation of the lunging (semi-prone) quad stretch is to grab the back foot with the opposite side arm. Start with your back-leg hand on the floor.
To deepen the stretch, go down onto your elbow.
One other set of quad stretches uses the wall. The idea is to position the knee of one leg at the base of a wall with the shin against the wall. The other leg can be in a lunge (though you can work towards a variation of the splits from here).
One variation is to use the front leg to push the hips towards the wall. From there you can also push your torso towards the wall.
A key point is to activate the foot and ankle of the leg you are stretching. Additionally activate the quad as if to resist your pelvis moving towards the wall. And, as before, lengthen your spine and create an upwards pull on ASIC and pubic bone.
A variation from there is to keep the torso upright but sink the hips forwards. (This is similiar to couch stretch.)This tends to shift the focus to the rectus femoris portion of the quadriceps.
From here you could go into a low-lunge-like position, placing hands and then elbows on the floor to stretch the hip extensors of the opposite leg. But you could then also straighten the front leg and move towards splits, again with the torso upright.
Because quad stretches, particularly when the rectus femoris is stretched, can stress the lower back, one way to relieve the lower back is to bend the spine forwards. While upright you can do this by dropping the sacrum and then from there dropping the sternum to round the spine forwards.
You can experiment with letting the ribcage sink down or actively pulling it down via the use of your abs.
Additional counterposing can involve bending forwards at the hips with one or both knees straight.