González was a product of the team's youth academy. He won the U-20 Liga MX Championship four times ranging from the Apertura 2010 to the Clausura 2012.[4]
González was selected as América's third-choice goalkeeper during the Apertura 2011 tournament, behind Armando Navarrete and Leonín Pineda. After a disappointing tournament and various movements in the transfer market — including the departures of Navarrete and Pineda — González was promoted to second-choice goalkeeper, this time behind new signing Moisés Muñoz for the Clausura 2012. He was handed the No. 1 shirt (Muñoz elected to use the No. 23). González would make his professional debut for América on 11 February 2012, coming on as a substitute for the injured Muñoz in the 64th minute against Atlas, with the match ending in a 1–1 draw.
After a car crash which involved Moisés Muñoz, González was expected to start for América for the Apertura 2012 tournament. He would also be América's starting goalkeeper for the Copa MX tournament, playing in every group-stage match, as well as playing in the following Clausura edition of the Copa MX.[5]
After poor performances in March 2016 from Muñoz, manager Ignacio Ambríz named González as starting goalkeeper, earning his first non injury-related league start on 13 March in a 2–1 victory over Guadalajara.[6]
González received his first call up to the senior national team on 31 January 2017 under Juan Carlos Osorio for a friendly match against Iceland.[12] He wouldn't make his national debut with the senior national team until 11 September 2018 in a match against the United States, where Mexico lost 0–1.[13]