Joe Cunningham: three home runs in first two games.

Joe Cunningham is the only player to hit three home runs in his first two games. He broke in with the St. Louis Cardinals in 1954.

Baseball Digest, page 67, May 1960 [Public domain], via Wikimedia Commons

Wednesday, June 30, 1954, , Crosley Field

Attendance: 12,021, Time of Game: 2:57

Cardinals 11, Reds 3
HR: J Cunningham (1, off Art Fowler, 5th inn, 2 on, 2 outs to RF)

Cunningham was 2 for 5 with 5 RBI and played first base.
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Thursday, July 1, 1954, , County Stadium

Attendance: 19,791, Time of Game: 2:24

Cardinals 9, Braves 2
HR: J Cunningham 2 (3, 2 off Warren Spahn, 2nd inn, 0 on, 0 outs to Deep RF; 3rd inn, 2 on, 1 out to Deep RF)

2 for 3, plus BB; 4 RBI.  Hall of Fame lefty Spahn walked Cunningham after the two homers, then got him to ground out P-1B. Spahn pitched 7 innings despite being hammered.
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Cunningham did not homer in his next four games but he tops Aaron Judge hitting a home run in each of his first two games this weekend, starting with:

Fritz Brickell and Bobby Richardson hit their first home runs in the same game in 1959, a bit like Tyler Austin and Aaron Judge yesterday. Sunday, August 14, 2016

... in their first plate appearances (PA) Austin and Judge homered, back-to-back jacks. Supposedly it's the first time in history that teammates homered in their first PA in the same game.
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Five players have hit two home runs in their first game:
Bob Nieman 1951
Bert Campaneris 1964
Mark Quinn 1999
J. P. Arencibia 2010
Yasmani Grandal 2012