Tough gig. String quartet music is intimate and tends to emphasize the interplay between musicians and the subtleties of their musicianship, and was traditionally intended for small audiences. They tend to be quite low in output level and traditionally rely on the performance space for both acoustic reinforcement and 'verb, and even with help from a good room they tend to sound rather quiet and distant in a big hall (it's "chamber music" after all). With no room to speak of the PA will need to provide both reinforcement and 'verb lacking from no room, and the presence of wind complicates everything. Sweetening the PA with some reverb will be advantageous (a subtle but large-room long tail may actually be appropriate at the canyon edge!).
Stage arrangement of musicians is most likely to be setup in an arc, either in the traditional placement: first violin, second violin, viola, cello (left to right as viewed from the audience), or the more modern configuration with cello and viola positions swapped, as that positions the cello facing the audience for a bit more acoustic projection of bass.
I've had good results recording acoustic string quartet performances in a very mediocre room with an ORTF pair placed close, low and in front, down near the floor angled upward toward the second violin in back. The "from below" angle allows the stereo pair to be placed relatively close to the ensemble, which helps gets good level, reduces bad room contribution (not an issue outdoors), improves audience sight-lines compared to more even with the viols or above, and helps keep the violin sound warmer and less strident than it tends to be from a relatively close position above.
For the PA mic'ing I'm not as much help- you may need to resort to close mic'ing each instrument. However, an omni dead center of the arc of musicians placed at chest height might actually work well (traditional bluegrass style), as that would provide relatively close equal distance to all 4 instruments and minimal susceptibility to wind noise. Maybe not though as quart string fiddlers won't be re-positioning and leaning in to the mic. Probably need to keep the PA cabinets out in front sufficiently to maintain sufficient gain before feedback