My PCP's office has used Quest for as long as I have been a patient there (20+ years), even though the doctor group is associated with Houston Methodist Hospital System. I have never had any issues with Quest; it is pretty seamless since the doctor's office usually submits the insurance claims.
MD Anderson has its own testing labs so they can provide almost immediate results. During chemo, I used to go in for labs before seeing my oncologist, and by the time I got to his office on another floor, he had those results to review with me.
Same with scans, biopsies, etc. When I went for the initial diagnostic mammogram, sonogram, and needle biopsy, I was advised that I had cancer that same day before I left the facility.