I would recommend - strongly - not to think of it as a 'baby' by any stretch of the imagination. It may not be fully grown, but it has the capacity straight out of the egg to kill damn near any untrained, unarmed metahuman it happens to be next to. Don't ever make the mistake that it's going to be totally ignorant, etc. etc. - remember that the Great Dragon is training it via telepathy from very early on.
I would also suggest - strongly - that the egg be one in Hestaby's care, but not one of Hestaby's. (She hasn't mated in the 6th World - at least not yet.) Shadowrun dragons do not actually care for their own eggs; they turn them over to a 'step-parent', specifically one of the Greats, to be cared for, trained, etc. It might actually be one of the Sea Dragon's, depending on when you're playing; Rhonabwy (I think - one of the Welsh ones) conned the Sea Dragon into breeding, then swiped the resultant egg(s) and turned them over to Hestaby. Which means if there's someone holding the egg hostage, they might be trying to leverage against both Hestaby and the Sea Dragon.
Otherwise, the egg might have possibly come from one of the other Greats (Masaru is known to have had an egg - part of an adventure back in the day), but would more likely be an egg from one of Hestaby's group of associated adult dragons. Great Dragons seem to tend to form a 'coterie' of adults, both to do their bidding, to act as lieutenants, and to be students to boot.
As an aside - recalling Shadowrun-canonical Earthdawn - there were adult dragons who tried to raise their own eggs. As I recall, the resultant children were ... troublesome.