In real life the main country club in Hartford is the Hartford Golf Club in West Hartford. It does look like the sort of place where Richard Gilmore would feel at home.
It doesn’t have a rose garden, which Emily told Richard to be sure to show Rory at the club. However, Elizabeth Park in West Hartford has a famous historic rose garden and is less than ten minutes drive from the golf club. While this probably wasn’t what the scriptwriter had in mind, in real life it would be possible to take Rory golfing and to see a rose garden in the same morning.
Exterior shots of the golf club were filmed at Brookfield Farms in Thousand Oaks, California – the same place exteriors of the Independence Inn were filmed.