No, Downtown Disney Guest Relations does not sell tickets for miniature golf. You must buy them at the specific miniature golf location at either Fantasia Gardens (next to the Swan and Dolphin) or Winter/Summerland (next to Blizzard Beach).
Which course would you guys recommend to be the funnest???? And about how long does it take to play a round of golf for a family of 4 (2 adults, 2 kids under 7)