Garfield was selected as a compromise candidate at the 1880 Republican National Convention, when delegates deadlocked between John Sherman (who Garfield supported), Ulysses S. Grant and James G. Blaine. According to Candace Millard's Destiny of the Republic, Garfield was shocked when delegates started voting for him, and although it's been awhile since I read the book, I believe Millard writes that he even tried to convince some of them not to do so. This story is of a piece with the general theme of the book, which portrays Garfield as an honorable man who was running for President not because he personally wanted to be President, but out of loyalty to his party and a desire to do right by the country.
I don't exactly disbelieve this, but I do kind of doubt it, because it seems like the kind of thing that might grow up around someone like Garfield who met a tragic end. It just seems to fit very neatly with what people at the time believed a leader should be like. Millard also describes how Garfield didn't personally campaign for his own election, but left it to surrogates to speak on his behalf, because the cultural norm at the time was for aspirants to higher office to appear disinterested in it - to cultivate the idea that they were like a Cincinnatus or a George Washington, someone who was acting out of duty rather than personal ambition. Garfield's story as recounted in Millard is almost too good to be true, when it comes to fitting into that idea or political norm - he didn't want to do it, he just did it out of loyalty, and in fact was initially horrified that delegates were voting for him because he didn't want to betray Sherman. So my question is this: is that narrative correct? As best we can determine today, was Garfield really just acting out of a sense of honor and duty, or did he have a personal ambition to be President as well?
Well, this sort of thing is complicated. It is true that when 16 out of the 20 Wisconsin delegates shifted their votes to Garfield, he attempted to raise a point of order objection that he did not consent to receive votes, but since the convention had been dragging on, and it was clear that a compromise candidate between the Grant men and the Blaine men was needed, the Chairman, George Frisbie Hoar, ruled him out of order and later stated that he did so, because he did not want to stop the momentum for a compromise candidacy by the candidate. What really put Garfield over was the swift decision by Blaine to come to the conclusion that backing Garfield was preferable to letting Roscoe Conkling (Grant's convention manager) outmaneuver him in nominating a compromise candidate that would be more acceptable to Grant (since by that point it had become obvious that none of the three leaders going into the convention: Grant, Blaine, and Sherman were going to be nominated). Sherman took a similar tack, advising the shift of all his delegates to Garfield to, "Save the Republican Party".
While Garfield may not have wanted to be a candidate and he probably was shocked at the result (he was described at the convention looking pale and half-conscious, today we might call this looking like a zombie), it's fairly clear that he wanted it. He sort of gives it away in his letter to his wife Lucretia where he wrote, "If the results meet your approval, I shall be content with the nomination". She was absolutely thrilled and quickly approved. He resigned his Senate seat that he'd just been elected to and did everything that a nominee for the Presidency at that time would do. He conducted the traditional front porch campaign where visitors called on him and his surrogates did the dirty work.
There is nothing to suggest that after being nominated that he really didn't want to be President. He could have refused (as Levi P. Morton did at that same convention when he was asked to put forward his name for the VP slot) outright, but he did not. His objection was fairly perfunctory. It's also interesting you bring up George Washington, because he was quite ambitious for the top jobs himself, but he learned very early that the best way to be ambitious was to act like you weren't. I suspect Garfield was much the same.
My primary source is the comprehensive biography of Garfield, Dark Horse: The Surprise Election and Political Murder of James A. Garfield by Kenneth Ackerman.