Here is a solution for breeding a gold quintar with an initially empty stable:

obtain RIVER WOKE from world
obtain BLUE TRUSTY from world
breed RIVER WOKE with BLUE TRUSTY (making BLUE FANCY)
obtain RED WOKE from world
breed BLUE FANCY with RED WOKE (making HIGHLAND BRUTISH)
breed BLUE FANCY with HIGHLAND BRUTISH (making HIGHLAND FANCY)
breed RIVER WOKE with HIGHLAND FANCY (making AQUA BRUTISH)
breed RIVER WOKE with AQUA BRUTISH (making AQUA WOKE)
release BLUE TRUSTY
obtain DESERT BRUTISH from world
release BLUE FANCY
breed DESERT BRUTISH with HIGHLAND FANCY (making DESERT FANCY)
release DESERT BRUTISH
breed RIVER WOKE with DESERT FANCY (making BLACK BRUTISH)
release RIVER WOKE
breed BLACK BRUTISH with DESERT FANCY (making BLACK FANCY)
release BLACK BRUTISH
breed BLACK FANCY with AQUA WOKE (making GOLD BRUTISH)