The simple answer is that, it all depends. You are entering areas of interstate laws concerning which state can hear custody questions (the UCCJEA).
There are also a number of fact questions, such as how long you have been living in Utah; did you intend to make Utah your residence; and where the child has been living over the past several months.
Is there any good reason why you have not started the divorce process?
The best thing to do is find an attorney who has experience with Utah's version of the UCCJEA, and work with that person. In that way, you may be able to achieve your goal, though you may have to return to Utah for any trial occuring in the divorce.
Sure the father may fight you, but at least you might possibly obtain support orders, and the ability to relocate.
Answered on May 14th, 2015 at 7:19 AM