Kobolds do lay eggs, which mature for 2-3 months before hatching
Volo's Guide to Monsters features an entire Monster Lore section about kobolds, including a brief examination of their biological life-cycle. Of their birth and ageing, it says that:
Kobolds grow and mature much more swiftly than members of other humanoid races. At 6 years old a kobold is considered an adult. [...] A female can lay up to six eggs per year, and an egg matures for two to three months before it hatches.
This is roughly congruent with lore from previous editions of the game, too. The 3.5e Races of the Dragon also describes kobolds in some detail and states that:
As reptiles, kobolds are hatched from hard-shelled eggs. Once a female kobold has been fertilized, she lays one egg within two weeks, with a 10% chance of laying two eggs. The egg must be incubated for 60 days, after which time it hatches into a kobold wyrmling that is able to walk and feed after only a few hours.
Based on this edition, if you had a relatively mature kobold egg already, your plan might still have been viable; Races of the Dragon notes that an egg prematurely cracked in the final 15 days of its maturation still produces a viable kobold wyrmling, albeit one that might mature a little more slowly than their fellows.