You Got Problems, Kid: True Polymorph creates creatures of CR 9 or less - this is a mondo-powerful spell! But for creature-into-creature casting, the CR prime-resultant must be less than the CR of origin. Simply put, this adventure's premise pends on a Green dragon egg having less Challenge Rating than this 1/4 CR Urd. Also of interest: polymorph-spells keep the 'alignment and personality' of the origin-creature. Do memories, culture &/or hopes-dreams survive this process? Hence the weird title, above.
True Question Asked / Answered: 'True' dragons are specifically the ten metallic-colour breeds or races listed in the monster manual. Other creatures of the dragon-type do not count as 'true': Pseudo dragons, wyverns, guard drakes and dragon turtles (and many others) do not count even if they have some or all of the True traits. For example, Faerie dragons have language, flight, spells, breath weapons and even age categories - but are NOT true. Some may point out that this urd could easily have become a Pseudo dragon - or virtually any other dragon if she just attained enough class levels first. Cut that out! I want her to become a RAW Green egg. We may add ham later on, if you like that sort of thing.
Which came first? The egg, of course: Assuming the egg is less than CR 1/4 (and this actually 'works'), what would the kobold remember of itself through this process? Most foetus-stage creatures would (logically) remember NOTHING as they are Tabula Rasa or blank-slate. However, dragons are smart - once born, Green dragons have 14 intelligence and speak Draconic. This implies vast A Priori or vast pre-set knowledge. With that intelligence and wisdom she could explain many key philosophies and even do quantum homework - albeit lacking adequate citations & references (i.e. not survive a day at Stack Exchange).
Would a foetus that keeps 'alignment and personality' include 'memory'?
Why ask / why bother / so what?? This previous life-memory changes the whole adventure! If this Green dragon remembers being a kobold, she KNOWS she is the product of a simple (albeit powerful) spell and is not a Green dragon. She is a true dragon via true polymorph that is not truly a dragon at all (possibly ironic). Also, if she does NOT remember, she would require either TrueSight magic (??) or some Legend Lore (??) or some other plot-mechanic to discover her previous life-form.
In any event:
Her survival as a Green dragon (form) pends on avoiding various Dispel magics - as well as avoiding dropping to zero hit-points.
A final vexing question: Both kobolds and urds only live until 135 years maximum. An ancient Green dragon is easily 801+ years old. Does the creature 'inside' the polymorph age at the same pace, similar to Willow's friend? Or does her origin-format NOT age... in a similar way to, um... a petrified person might (not)?
This part may be six more StackExchange questions...
I cannot find it, but it would help to know what forms of dispels &/or anti-magic temporarily or permanently remove her form. And what magic(s) work to discover her dispel-vulnerability &/or history.
Can she survive (as her dragon-format) within a Dispel Magic spell in her general area? What if it is (successfully) cast to remove a spell or curse on her, would her polymorph also drop? Does this spell roll its 'saving throw' separately?
What if she enters an Anti-Magic Shell - or gets hit by the central eye of a (normal) Beholder? Does she become a bold yet old kobold whilst in this non-magic only to pop back into dragon shape once she is out? If so, a Beholder could / would / should really make use of this (i.e. 'plot relevant information').
What does True Sight style magics reveal whilst she is in Green dragon format? Do they see that bold-old kobold walking around inside her, mimicking her actions? What does that Lore of Legends say? What would Identify reveal?
Once upon a time a simple urd-kobold was abandoned due to her wings and 'gift' of flight. Yet, here we are at Stack Exchange - sorting out her inner-most identity problems. Sorry for the mess / hope you enjoyed reading this as much as i enjoy writing it.