Quadraphonic L P played with a 2 chan. cart.

Yes.
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One kind of quad LP (SQ encoded) is simply a matrix system where the 2-channel signals are processed to derive four signals. No problem for the cartridge. It will probably sound OK.
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Another kind of quad LP (which I never tried) requires a cartridge with HF response up to at least 40 KHz, because the signal that separates out the rear channels is carried by FM modulation of a carrier in the range of 20 to 40 KHz. (Note that the cartridge does not need to have flat or low distortion in this upper range because the modulation is FM). If you play such a record with a cartridge lacking HF response you might damage the grooves. However, the good news is that because of this quad system the HF response of phono cartridges made a great improvement, and most if not all cartridges sold today would work fine. The quad system died, but we still benefit from it.
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This thread brings to mind one of the first foray's into stereo I remember. It was a tonearm that became 2 arms at the cartridge end. You put in 2 monophonic cartridges and it used special records - one channel on the 1st half of the LP, and the other channel on the other 1/2. The arm tracked both channels at the same time. It looked like a snake with 2 heads. I believe the arm and the records were manufactured by Cook.

I still use a Lafayette SQ quadrafonic decoder for the few QS records that I have. I also use it for a way to 'simulate' rear channel information for ambiance effect. There has been no damage done to the original SQ 4-ch recordings by using an 'ordinary' 2-ch phono cartridge - in my case, either the B&O MMC3 or Sumiko Bluepoint special.

Fear not, play away.

Bob P.

Eldarford, I believe the quad LP system, which uses a FM signal, to which you refer, was the JVC CD-4. It was arguably better than the SQ & QS systems, but at least, as you said, it improved the high frequency tracking of cartridges to this day.

Inpepinnovations...I also used a Lafayette SQ-L decoder, which is now in my collection of defunct audio stuff. It was probably the best of all the logic decoders. Mine is all discrete components and must have cost much more to build than the $99 I paid for it when it was discontinued. My guess is that the circuit was a prototype that they intended to implement in ICs. I modified it a bit and used it as my preamp for several years. It had a bit of gain which made this practical.

To all that responded or interacted to my initial inquirey:

Thanks; I am always amazed at the level of knowledge and interaction amongst the members here on Audiogon. Although, I am relatively new as an audiogon member and a novice to the level of information on the subject of audio equipment and its theory which the body of menbers have attained; I hold myself out as a 70's student who is willing to learn.

Thanks,

Bruce