The song on the jukebox while Howard and Alice are dancing is 1939s "Brazil."
Andy mentions going to Chimney Rock. Chimney Rock is located in the North Carolina Appalachian Mountains. 24 miles southeast of Asheville, North Carolina.
Based on the relative ages of Andy and Opie, Alice would have had to have dated Andy in their 20s: somewhere between 1944 and roughly 1950. These were the years between Andy's breaking up with Sharon De Spain at graduation, and his courtship and eventual marriage at age 28.
Andy reminds Goober that Helen comes from Kansas, although Goober tries to plant the suspicion that maybe she's not. In the next season her life in Kansas becomes a major issue in "Helen's Past."
Judging from the color of the car that Andy and Helen are arguing in, it was Howard's they all drove up to the cabin in.