Gregorio was born in Iurreta, Bizkaia on 24 December 1899 to Juan José Alberdi and Genara Bolinaga. His parents were farmers. Juan José passed away before Gregorio decided to emigrate to the United States to join his brother and uncle, but Genara was supportive of her son's decision. She gave her son the money he would need for his passage but told him to come home soon.
In 1919, at the age of 20, Gregorio boarded the "Chicago" in Bordeaux, France, and crossed the Atlantic. The ship made a stop in Habana, Cuba before continuing on to New York, where Valentin Aguirre was waiting to take Basque immigrants to his boarding house. Aguirre helped Gregorio find the train that would eventually take him to Boise, Idaho. A man named "Barbero" met Gregorio and his group at the Boise train depot and took them to his boarding house. Gregorio worked as a sheep herder in Kuna, Idaho until 1925, when he went to work at a lumber camp near Centerville. As time passed he missed family in Iurreta, but a shallow bank account and news of war in Spain kept him from visiting them.
Gregorio stayed in the US and met his wife, Francisca Egurrola, who had come from Lekeitio to visit her sister. They married in Reno, Nevada, but settled in Caldwell, Idaho, where Gregorio and his brother-in-law, Cristobal Gabiola, had established the Gabiola & Alberdi Cattle and Sheep Company in 1927.