(Guide may be outdated as i do not play anymore)
STEP 1: GO TO PLANET "VENUS" > KILL JACKAL BOSS ON FOSSA > this will drop you what u need to start quest.
STEP 2: GO TO CODEX , this is located when u press ESC (bringing up the quick bar) then PROFILE > CODEX > QUESTS, here you will start your quest!
STEP 3: THE QUEST TELLS YOU TO GET THE INCUBATOR > This will be from a survival mission at jupiter called elara ! complete 5 mins and extract with the incubator!
STEP 4: GO TO MARKET > EQUIPMENT >COMPONENTS > Now you must buy what is required for kubrow making, "Incubator Power Core/Blueprint" , DNA Stabilizer Kit, Genetic Code Template/Blueprint. Once you have these blueprint build them in your foundry > comes under GEAR <>
STEP 5: THE PLANET EARTH ON >"E PRIME" WILL BE AN EXTERMINATION MISSION , On this you will find kubrow dens and kubrows! DENS WILL DROP EGGS, KUBROWS DROP MODS (mods for kubrow) Eggs are very rare so keep destroying dens , this may take some time.
STEP 6: NOW YOU ARE SET TO BREED THE PUPPY GOOD LUCK TENNO!