If they're dominant enough, I don't see why not.
He didn't really come close to winning it, but Mariano did finish 3rd in the Cy Young voting in 1996, the year before he became the Yankees closer.
Another interesting tidbit looking at Mariano's record. The year he came in 2nd place for the Cy Young as a closer in 2005, the award was won by Bartolo Colon, who happened to win 21 games for the Angels. Johan Santana finished 3rd while pitching more innings then Colon, and having a superior season in just about every other way. In today's climate he likely wins the Cy Young going away.
He should arguably have 3 Cy Young awards in his pocket...along with a pitching triple crown, a Gold Glove, 3 ERA titles, 3 Strikeout Titles, 4 WHIP titles, and a whole bunch of other bolded black ink marks on his baseball reference page. Many, many more then Degrom.
Santana pitched about 800 innings more then Degrom has at this point, and his career was deemed too short apparently, by the HOF voters. Santana was kicked off the ballot after his first year of eligibility.
I mention this because Santana seems the closest comparison career wise to me regarding Degraom, even though they had different paths to get where they got to during their careers.
Both electric, both great pitchers, and both will eventually be thought of more highly then many pitchers who will get into or are already in the HOF.