Rules for something as specific as this are unlikely to appear in D&D 5th edition, so as the previous answer correctly notes, this is up to the DM.
There is a little lore on this topic in the earlier D&D 3rd edition, although it only partially answers your question.
Celestial animals appear to be able to reproduce. According to Races of Faerun p.112:
Aasimar are the descendants of humans and some good outsider, such as a true celestial, a celestial creature, couatl, lillend, or even a servant or avatar of a good deity.
"Celestial creature" here refers to a template applied to a creature, often to create the stats for a celestial animal, but can also be applied to a humanoid (e.g. according to Complete Divine p.110, Corellon has celestial elves in his service). As per Manual of the Planes, celestial animals are often created from the souls of dead humanoids arriving in the Beastlands, and perhaps also from the souls of animals.
Races of Faerun p.112 describes the result of such interbreeding:
The child of an outsider and another creature is a half-fiend, half-celestial, or half-elemental.
By this definition, a celestial dog plus a regular dog produces a half-celestial dog, which would be more powerful than either of its parents. However, as per the 3.5 Monster Manual, half-fiend and half-celestial templates are applied only to creatures with an Intelligence score of 4 or higher, which rules out normal animals.
Half-fey, Fiend Folio p.89, have no such limit. They are strange versions of normal D&D creatures with butterfly wings and resistance to enchantment.
However, I suspect that the intent of the half-celestial, half-fiend, and half-fey templates in D&D 3e are for the likes of more powerful outsiders interbreeding with humanoids, rather than celestial animals with other animals, since the latter creates the paradoxical situation where the offspring is considerably more powerful than its original form.
When you consider the actual power level of a celestial animal as a 5e familiar, it has no innate power above a normal creature of its type, aside from its connection as a familiar. It seems unlikely that the cross-breeding of such a celestial ferret with a regular ferret would produce powerful offspring. What I'd expect is more like what Races of Faerun p.112 notes happens when e.g. an aasimar breeds with a human, which is a 50-50 chance of being either an aasimar or a normal human, with the possibility of the celestial bloodline remaining dormant and re-surfacing generations later.
In short: D&D 5e has no rules on familar reproduction, and while D&D 3e lore suggests that celestial, fiendish and fey creatures can indeed interbreed with regular creatures, the exact nature of their offspring is unclear, particularly as D&D 5e's celestial/fiendish animal familiars are weaker than their 5e counterparts.