I'll put forward a very ruthless option.

Raising children has the biological time constraint vs raising a dog (has no biological time constraint)

There should the sequence of priority in life. If the dog(pet) has higher priority than the husband or kids, this kind of woman should be left to spend their life with their dog. Let her remember the divorce in her life is caused by a dog. A simple and straight forward option. There is no need to bring other element into discussion.

After the sequence of priority is correct, raising the children has taken the place, both of you have more leisure of time for the pet(dog) in future, that time it is still not too late let her have the dog again.

Hence, in between the dog & divorce - let the wife decide on the agreed date. If she choose the dog on the dateline, execute the divorce procedure without emotion. If she choose husband & kids over the dog, find the dog a new owner.

She grown in a small happy family, her parents and siblings are all also well connected.  I would say she grown in happy and steady family.

One thing i do not understand is - she never owned a pet (dog) since she was kid and when we were dating for almost 7 years or so, we never thought of owning a pet also.  If you grown with family that have pet dog, i can understand and definitely i will not going into marriage with her, cos im not a pet person.

And when we talked, she said she only realised she loved dog so much after she got one. 

Speechless.

Its not too late but u guys should do fertility test (both of u).

Imagine if u divorce for not wanting kids but one or both of you actually hv fertility problems.

Maybe this should be a first step.

seriously imho, it has nothing to do with the dog.
I think your wife is not really ready to become a mom. Because you can have a dog and have a kid together as well. Try go check out lots of videos in Youtube where a couple with existing dog welcome a child and the dog even become good buddy of the child and very protective of the child too. (maybe you can consider showing such video to your wife too)...
but take a way the dog or put aside the dog, i think without the dog too your wife most likely will give you the same answer that she's not ready to become a mom...

Taking care of a child is way way way way way more complicated and harder than taking care of a dog... i'm speaking from my own experience...i have dog...i also have kids... 
good luck buddy...

on a separate note....you love your wife and you made a vow to marry her... just because she don't want to have kids you want to divorce her? you love your wife or you love kids? again, just my own opinion, I love my wife first and whatever after that, both should discuss and come to mutual agreement together. In this case, if my wife is not ready to have kids, i shall stick with her decision as well until she is ready...even if it means there is a possibility we won't have kids...