The quadriceps are affected by the hip joint. In part this is because of the rectus femoris, but it may also be due to other connective tissue connections which aren't mentioned in a lot of simple anatomy texts.
To take the hip joint out of the equation when stretching the quads, bend the hips forwards.
A simple way to stretch one quad at a time is to have one knee in hero position. In hero position you sit with your butt to the inside of the heel. The other knee is bent with the foot against the inside of the hero leg.
Have the butt of your non-hero leg on the floor.
Lean to the non-hero side. To stretch the quadriceps in this position slowly sit upright so that the weight of your body pushes down on your hero leg buttock. Then slowly relax.
To add more weight to this quadriceps stretch do a side bend towads the hero leg side. To add more weight, reach one arm, and then the other arm to the side.
For leverage you can press your other knee into the floor.
Move slowly and stop at any point you feel knee pain. If you experience knee pain first try pressing the toes of your hero leg into the floor.
Another Quad stretch is half hero pose. This is a variation of recling hero or supta virasana.
In half hero pose, one leg is straight. The other is folded back at the knee with the shin to the outside of the thigh and the top of the foot on the floor.
If you are new to these types of kneeling stretches, focus first on sitting upright in this pose.
Gradually lengthen your spine upwards. Interlace yoru fingers and press your palms up to the ceiling. Then slowly relax. As you lengthen use your body weight to gently push your hip on the hero leg side down. Then gently relax.
Once you've done both sides you can then work on Full Hero Pose.
In this variation of hero pose, simply sit upright.
If your butt isn't touching the floor then you can work towards that in the same way as in the kneeling variation above. Lean forwards a little, then slowly sit upright to help your hips sink down. Lean forwards again and repeat.
Another way to gently add weight and reduce it is to use your arms. Press your hands down into the floor initially, then gently relax your arms and lift them to add weight to the quad stretch. Then to relax the stretch press your hands back down into the floor.
If your butt is only a few inches off of the floor then you can try holding the position. Slowly lengthen your spine and then relax your spine. As you lengthen your spine press your toes down. Relax your toes as you relax your spine.
The next series of quad stretches adds the hip flexors to the stretch.
This makes the stretch slightly deeper. For those interested in meridian stretching, this stretches the stomach meridian along the front of the hip, the front of the thigh and the front of the knee. And if you make the toes active it can also stretch the stomach meridian along the top of the foot.
Once you can get your butt to the floor (or a few inches away from the floor) in either hero pose or half hero, you can try reclining half hero pose.
As with getting your butt to your heels and then to the floor, work towards this quad stretch gradually.
Start with your hands behind you. Then sink to your elbows.
You could also try to get the crown of your head to the floor with your spine bent backwards. From there you can then work on getting the back of your ribcage to the floor.
Work on reaching both legs forwards. Focus on pushing the knees forwards away from the pelvis.
And here again press the toes of the hero foot into the floor.
You may find that the psoas pulls on your lumbar spine. Focus your awareness on your psoas. You may find that you can get it to relax.
Alternatively, focus on feeling your lumbar spine.
Gradually work at pulling the back of your ribcage away from your pelvis. You can reach your arms back also.
For a change in this quad stretch, bend your other knee, place the foot on the floor and lift the pelvis. Continue to push your hero knee into the floor. Work at keeping your pelvis level from left to right.
For an even more sensational yoga quad stretch, hug the lifted knee to the chest.
It may be next to impossible, but try to hug the knee to the chest with the knee straight. I cheat and prop one elbow on the floor to do this. I also have experimented with starting with the leg out to the side and then lifting it.
A similiar quad stretch involves using the wall. This offers the advantage that you aren't subject to gravity. Instead you provide the stretching power to stretch your quadriceps.
Start in a lunging position with the back knee against the wall and the shin of that foot up the wall. You can use a folded up mat or towel beneath your knee and even between your foot and the wall.
To stretch the quad in this position use your front leg to push your pelvis back, towards the wall.
As you do so, lengthen your spine upwards.
You can push your hands against your knee or reach them, one at a time, upwards, against the wall.
Try to position your foot so that you can get your pelvis to the wall. Note how in the second picture my lifted foot is to the outside of my buttock in hero position.
A variation is to place your elbows on the floor. Focus on sinking your hips. At the same time press your foot into the wall.
For this next combined hip flexor quad stretch you might want to start off with a low lunge.
In low lunge focus on sinking the pelvis down.
Note, if your knee is uncomfortable then place towel under your knee or fold up your matt for comfort.
Bend your back knee and grab your ankle with your front leg hand. (Use a towel or folded over yoga mat under your back knee.) To make it easier to grab your ankle, lift your pelvis.
Start with your back leg arm straight.
Slowly sink your pelvis down while pulling the foot forwards with your arm. Turn your ribcage towards your front leg.
To deepen the stretch rest your back leg elbow on the floor. Again, sink your pelvis, pull the foot forwards. Turn towards your front leg.
(I forgot to mention. Have some tissues near by.)
If you get knee pain in this position, try activating the foot and ankle of the back leg. Make the foot and ankle feel strong.
For extra bonus points, lengthen your spine. Lengthen your neck.
To stretch the quads on both legs at the same time, and the hip flexors also, you can work towards reclining hero pose.
A slightly more challenging stretch for the quads that also exercises the arm is frog stretch.
When doing this pose one leg at a time, use your supporting arm to press your ribcage higher. If you can get your fingers to point forwards (the hand that is on the foot), then press your foot upwards and your hand downwards at the same time. Resist the stretch. Then relax the quad and let your hand push your foot down beside your hip.
The hand action is like you are doing a push up or cobra pose. As you push down with your hand try to lift your chest up.
You can also do both legs at the same time in this pose.
Use your spinal erectors to bend your spine backwards. You may find it helps at first to push your feet upwards against your hands. You can then use your arms to lift your ribcage. From there, with ribcage lifted, then gradually reduce leg tension and use your arms to push your feet down towards the outside of the hips.
For fun you can combine quad stretching, hip stretching and balance in this balancing one legged quad stretch that also stretches the hamstrings. It's also a possible substitute for ardha badda padmasana.
You could start with the bend knee down. Then gradually lift the knee so that you add a hip flexor stretch to the quad stretch.