Is there a correlation between frequency response and sound quality?

It is not clear what the "right" frequency response for headphones should be, flat sounds odd to lots of people. For one thing a flat speaker in front of you at about 30% angle, as stereo speakers are commonly positioned, will not produce a flat curve at your eardrums. Your external ear adds peaks and valleys at certain frequencies and that is among the things your brain uses to determine direction.

In practice most high end headphones are distinctly not flat at the eardrum, because they often attempt to shape the sound curve to fool your ear so it hears a response much like it would hear from a flat stereo speaker.

A discussion you might find interesting in this context can be found here.

Well everyone's external ears differ from everyone elses, so they all affect sound differently to a certain extent. Thing is, your brain knows your ears very well, so a really successful adjustment would be different for everyone.

What works well for you might not work for me, which puts a mass manufacturer in a difficult situation.

Another problem is that headphones totally separate the sound from left and right, while with speakers it is mixed and each ear hears both speakers. Some headpbone amps compensate for this with a time delayed crossfeed, but again not everyone's head is the same width so this is only a crude approximation.

All in all designing a good headphone is even more of a black art than designing a loudspeaker where at least the most desireable curve is generally agreed upon.

It is all subjective. Some like to over-think.

Another good thing to read: Stereophile: Between the Ears: the art and science of measuring headphones

A perfectly flat frequency response of your headphones would make your ears bleed! Literally - too much highs, emphasized midrange and K 701's bass.

Something else to ponder,Do the size of the ears equate to a variation in sound/Do the angles of the ear equate to a variation in sound.
Not everyone is born the same.

To me, there is a big difference between frequency response and quality.
It is possible to have two headphones that are both undeniably excellent but have substantially different frequency responses. Conversely, it is possible to have two headphones of radically divergent quality with very similar frequency responses.
Quality to me is determined by things like clarity, detail, dynamics, and soundstage.

To a degree, we have to agree. But a neutral headphone should be likeable with music that is not coloured itself. You will often find hps with crazy frequency response that are likeable with some "unnatural" genres (like metal, electronica etc. ) but it is a different story. How can we then determine which is better?

I don't think Headphones can ever achieve a perfectly flat frequency response as well. There are resonant frequencies inherent in the 'covering' of the headphone cups and your ears. Some sort of an equalization effect has to be added to counteract this effect.

To me Knowledge and experience plays a key role in what my idea of a good sound quality is. Frequency response isn't as important to me as 'Coherency'. It is how well integrated these frequencies are that determines my idea of an headphones sound quality.

The referenced article is very interesting. I encourage everyone to read the article yourself, but to capsulize, Keith Howard (the author) did some reserch, measuring and listening. His reserch indicated that a headphone should not measure flat:

"If we are using headphones in place of speakers, then it might seem logical that the headphone's frequency response should imitate that of a sound source at head level, 30° off the median plane— ie, where the loudspeaker would be in a conventional stereo setup. In headphone parlance, this is termed the free-field or FF response assumption."

And showed what this response is generally considered to be.

But then comes a possible curve ball:

"Self-evident as the correctness of the FF-response assumption may seem, it came under concerted attack in the 1980s, principally through the work of Günther Theile at the Institut für Rundfunktechnik (IRT) in Germany (footnote 4). Using a Gestalt model of auditory perception, Theile argued that a free-field headphone frequency response would be appropriate only if the stereo image were perceived to be forward of the listener, as it is when reproduced over loudspeakers. As everyone who has used headphones knows, this is not the case—the image is generally perceived to be either inside or close around the head. Because of this, Theile claimed, a headphone with a free-field frequency response is perceived as spectrally colored."

Oops... this FF curve is different! So what's the bottom line?

"If you suppose that, as a result of Theile's work, there is now a headphone-industry consensus that the DF response assumption is the correct one, prepare yourself for disappointment. Headphones continue to espouse widely differing response philosophies: some close to FF, some close to DF, and others nearer to flat. Which is "right" remains a bone of contention"

Open ended is more like it. But to cut to the chase, Mr. Howard measured 4 phones, corrected them for both FF and DF curves, then auditioned them with an eye (ear?) to trying to confirm one or the other technique. His determination:

"I am beginning to suspect that a response somewhere between DF and flat is actually optimal. But these are early days; as other headphones pass through my lab, it will become clearer whether this is indeed the case."

The article shows various currently accepted curves, for example, the DF curve has a dip above 1 Khz level slightly below 3Khz, a peak at about 6.5Khz and a dip back down at 8Khz with a rising response above 10khz.

But, to answer the direct question, yes the frequency response matters quite a bit, but the kicker is there is no general acceptance as to what the measured frequency response should be! Things like harmonic distortion and dynamic linearity etc. also matter, so frequency response certainly doesn't tell the whole story in any case.