To begin breeding you need to beat Quintar "mini-bosses" that are located at specific places (usually in Overpass) and gain quintar-eggs of different types and natures.

Types:
Blue = Rental
Red = Normal
River
Desert
Highland
Black
Aqua
Golden

Natures:
Fiendish
Brutish
Woke
Fancy
Trusty

Ingame tips:
River and Trusty can only be found in the wild.
Fancy can only come from an egg.

Fiendish parent = Type and Nature equal to Fiendish parent.
Brutish parent = Type equal to Brutish parent, Nature from the other parent if possible.
Brutish will override Fiendish
Fancy parent = Allows mutations, meaning that the type can be something other than either parent.

In Rolling Quintar Fields:
Blue Fiendish

North West of Okinawa:
Red Trusty

West of Callisto Shrine:
Red Trusty

West of Northern Stretch:
River Woke

West of Catfish Sanctuary:
River Woke

North West of Dione Shrine:
Blue Trusty

North East in Quintar Reserve:
Red Woke

Nearby Yamagawa M.A.:
Blue Trusty

Poko Poko Desert South East:
Desert Brutish

Poko Poko Desert South West:
Desert Brutish

Poko Poko Desert North West:
Desert Fiendish

West of Ancient Labyrith:
Red Trusty

Nature:
Fiendish X Anything = Fiendish
Woke X Trusty = Fancy
Fancy X Trusty = Woke
Fancy X Woke = Brutish

Type:
Red X Blue and one parent Fancy = Highland
Fancy Desert X River = Black
Fancy Highland X River = Aqua
Black X Aqua = Golden

Breeding Requirements:
Mood must be "Happy!"
Required number of different race-tracks won for a parent if the other parent is of the following type:
Blue, red = 0
River, Desert, Highland = 2
Black = 3
Aqua = 4
Golden = 5? (not tested)

Example:
Desert X Black:
Desert parent must win on 3 different race tracks, the Black parent must win on 2 different race-tracks and both parents must be "Happy!" for the Breeding to be successful.

Recommended Quintars to Capture:
Blue Trusty
Red Woke
River Woke
Desert Brutish


Breed:
1: Blue Trusty X River Woke = Blue Fancy

2: Blue Fancy X Red Woke = Highland Brutish
Highland Brutish X Blue Fancy = Highland Fancy
Highland Fancy X River Woke = Aqua Brutish

3: Desert Brutish X River Woke = Desert Woke *
Aqua Brutish X Desert Woke = Aqua Woke

4: Desert Brutish X Blue Fancy = Desert Fancy *
Desert Fancy X River Woke = Black Brutish
Black Brutish X Desert Fancy = Black Fancy

5: Black Fancy X Aqua Woke = Golden Brutish


Note:
I picked Desert Woke and Desert Fancy as the quintars to breed into Black Brutish and Aqua Brutish because I found them the easiest to win 4 races with (1, 2, 3 and 4 or 5). If you like the Highland quintar better, you can change them to Highland Woke and Highland Fancy in the first part of step 3* and 4* respectively (Desert Brutish -> Highland Brutish). The breeding sequence in this guide was made when Black Quintar required the other parent to have won 4 races before it would agree to mate (now requires 3).

If you need help with the races there is a guide for it written by another member of the community.