May 1, 1984
The golfer in this game is a mustached man wearing a cap. While for player one his clothes are white with blue pants, when used by player two, his pants are black and all white parts in his clothes (as well as his golf ball) are changed to red, making him look somewhat like Mario, but it may be just a coincidence.
Apparently the american version of the game tried to get advantage of this similarity by placing the player two character on the cover (while most games use player one, like also the japanese version of Golf did). Since then there have been some debate between Mario fans if to consider him actually Mario or not, but he is never confirmed to be Mario in the game, manual or any other official source.
The only exception is Mario Mania Nintendo Players' Guide, released in 1990 and licensed by Nintendo of America, which included a number of games where Mario made small cameos, and among these is Golf (note that the screenshot provided actually shows the white-clothed Player 1 golfer). That guide, however, was written in America without any support from Nintendo of Japan, and took many liberties, such as stating that Mario's hat is hand-made, or using Pauline's and Donkey Kong's images from the non-canon Saturday Supercade animated show in their respective section, rather than from the games. For this reason the guide's statement is likely to be interpreted as the writer's opinion, not an official confirmation by Nintendo.
The 2008 game Captain Rainbow (see Captain Rainbow X Golf (Nintendo)) retconed the golfer to be named Ossan, finally specifying that he's not Mario, but still there are some Mario fans who like to consider Ossan the golfer in the japanese version and Mario the golfer in the U.S. version, or Ossan the player one golfer (white clothed) and Mario the player two golfer (red clothed).
Golf Japan Course
February 21, 1987
This time the golfer is clearly shown to be Mario on both the cover and in gameplay.
It should be noted that the game's graphics are the same of Golf, only with the golfer replaced by Mario. To some people this difference evidences even more that the original golfer was not Mario, but others believe it actually confirms that it was always meant to be him.
Also, the player two golfer is Luigi.
Golf US Course
June 14, 1987
A sequel to Japan Course, the golfers are once again Mario and Luigi.
Golf (Game Boy)
November 28, 1989
A remake of the NES game Golf, the golfer appears this time less detailed on screen, but it's once again a mustached man. It's hard to tell if he was intended to look like Mario, because while he shares some similarities with him, he looks nothing like the classic 8-bit Mario, and the original japanese box only shows some unnamed golfers.
On the other hand, the cover for the American and European releases clearly shows Mario, so this time we can consider the golfer to be actually him, at least in the international version.
Mario Open Golf
September 20, 1991
Mario Open Golf is the sequel to Family Computer Golf US Course, and is also a Mario game, thus being part of both series.
Mario and Luigi are once again the golfers, but this time princesses Peach and Daisy are also present as caddies. Notably, this is Daisy's first appearance after her debut in Super Mario Land.
Toad also makes a small appearance waving a flag if the ball lands out of a course bounds
NES Open Tournament Golf
September 29, 1991
Basically the western version of Mario Open Golf, while it features different courses and game modes and no longer has "Mario" in the title, it still features Mario, Luigi, Peach, Daisy and Toad in the same roles as the previous game.
A very small Monty Mole (an enemy from Super Mario World) also makes an appearance on the game's cover, but not in the game itself.
Copyrights for both series are held by Nintendo.