A 10gallon tank would be too small for adult goldfish to live in let alone breed in, breeding goldfish in an aquarium is unlikely to happen for the below reasons;
a. They need the right conditions to incourage them to breed, these conditions usually mimic the ones that come about in ponds and rivers when the seasons turn from winter into spring- in an aquarium these conditions are unlikely to happen by themselves.
b. Goldfish need plants (and lots of dense planting at that) to lay their eggs on, the problem is though is that goldfish also eat plants, particularly with common goldfish in the tank you are unlikely to have any plants survive and thrive for any decent ammount of time.
c. Goldfish will eat fish eggs, and after spawning has finished they will eat other goldfishes eggs (and even their own on occasion). In a large pond or river they are unlikely to find their eggs again after spawning, but in an aquarium they hardly have to look far to find the eggs.
d. Goldfish usually need to be 4-5years old on average to be mature enough to breed, most goldfish sold at shops are barely over 1 year old.
e. Goldfish will eat any fish smaller enough for them to swallow, so even if the eggs survive and hatch, any fry born in the aquarium are likely to be eaten by the goldfish.
Thus often the easiest way to breed goldfish is to have a pond of at least 500gallons and at least 3ft deep. For a pair of adult fancy goldfish i would advise at least a 30gal tank depending on the exact variety, or a 75gal for a non-fancy breed of goldfish.
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