Quigley and Nelson were at 7-under par through 13 and 14 holes, respectively, when rain forced each of the 78 players off the Greystone Golf and Country Club.
The field will return Saturday morning to complete the first round before the second round gets under way.
The 55-year-old Quigley made his record 178th consecutive start for events he has been eligible, breaking the mark set by Mike McCullough, whose run ended at last year's SAS Championship.
A six-time winner on the Senior Tour, Quigley began his streak at the BankBoston Classic in August 1997. It includes all six of his titles, 10 second-place finishes, 74 top-10s, 534 rounds, 39,552 strokes and more than $6.78 million in earnings.
Nelson was named Player of the Year in 2000 and won five times last year, but he is winless in nine events this season, finishing no better than fifth in any of them.
Forty players were under par, including defending champion Hale Irwin at 1-under through 12 holes. The 56-year-old has won twice this season and leads the money list with $951,000.
Jim Thorpe, who claimed the Tradition last week for his first major title, was even after 13 holes. He has claimed all five of his Senior Tour wins in the last 20 months.