Hi Ericdavidhoke, private outdoor space is hard to come by in town and is always in demand.
I am a dog owner myself, and while Hoboken is very dog friendly, more and more owners that are renting their homes for the first time, find themselves uncomfortable with the idea of having pets in their home with tenants.
You may have a better chance finding an apt in a large building that has a common yard. Hoboken does have some nice parks and designated dog areas. I would recommend you check some out.
Last, if you find an apartment that has a private yard but doesn't allow pets, you should make an appointment to see it, that is if the no pet rule is a landlord rule and not a building rule, and if you like the apartment offer to pay a non refundable pet fee of $500 or you could offer to pay more rent than the owner is asking. Hopefully one of these will sway them into renting to you, pending you have a good looking application.
Best of luck