The only way I know of to get a pet is to buy one. The pet shop in Qeynos or Freeport will sell you anything from a spider to a pig with lots of options including cats and a dog in between.
On the other hand, the baby dragons are a promotional item only available through purchase of the EQ2 collector's edition tin box set. This goes for between $70 & $90 depending on where you buy it and as of a week or so ago were still available in many stores.
You could also *upgrade* to the collectors version online, (for $90), and get the dragon but not all the other in box goodies, (like the cloth map, soundtrack CD, collector's gold coin, etc.)