Your attorney can get a stay from the Appellate Division if a trial judge refuses to do so. That doesn't mean he will get one, but it means he can ask. It depends on a variety of factors including the strength of the underlying decision, and whether irreparable harm will result from the delay (or whether it can be fixed with money) among others. Just because you don't get a stay, doesn't mean you won't ultimately win the appeal.
To specifically answer your question, the answer is yes.
Generally, a stay of enforcement of a judgment is found in NY CPLR 5519. Pursuant to CPLR 5519(6),
"6. the appellant or moving party is in possession or control of real property which the judgment or order directs be conveyed or delivered, and an undertaking in a sum fixed by the court of original instance is given that the appellant or moving party will not commit or suffer to be committed any waste and that if the judgment or order appealed from, or any part of it, is affirmed, or the appeal is dismissed, the appellant or moving party shall pay the value of the use and occupancy of such property, or the part of it as to which the judgment or order is affirmed, from the taking of the appeal until the delivery of possession of the property; if the judgment or order directs the sale of mortgaged property and the payment of any deficiency, the undertaking shall also provide that the appellant or moving party shall pay any such deficiency"
Your appeal concerns the possession of real property. Therefore, to stay the eviction pending appeal you will have to file a bond for an amount determined by the Court.
The short answer is yes. The specific facts and circumstances will determine whether in fact the appellate division will actually sign an order.