How we cite our quotes: (Paragraph.Sentence)
Quote #1
[…] my feelings do not permit me to suspend the deep acknowledgment of that debt of gratitude which I owe to my beloved country for the many honors it has conferred upon me; still more for the steadfast confidence with which it has supported me; and for the opportunities I have thence enjoyed of manifesting my inviolable attachment, by services faithful and persevering, though in usefulness unequal to my zeal. (6.1)
Washington gets personally patriotic here. It's like he's demonstrating his own love of the United States to show that he's just as into it as his audience. He owes as much to America as anyone, and he's willing to shout it from the rooftops. Metaphorically, of course—the address was printed in newspapers, so not a lot of rooftops involved.
Quote #2
[…] let it always be remembered to your praise, and as an instructive example in our annals, that under circumstances in which the passions, agitated in every direction, were liable to mislead, amidst appearances sometimes dubious, vicissitudes of fortune often discouraging, in situations in which not unfrequently want of success has countenanced the spirit of criticism, the constancy of your support was the essential prop of the efforts, and a guarantee of the plans by which they were effected. (6.2)
Some of the patriotism in the farewell address isn't as obvious as the waving of a flag on the Fourth of July. Washington credits the people of the country with a lot of his success—and since the American people are pretty essential to the country, celebrating them is really celebrating America. Otherwise, you're just really into the mountains and amber waves of grain.
Quote #3
The name of American, which belongs to you in your national capacity, must always exalt the just pride of patriotism more than any appellation derived from local discriminations. (10.3)
Well, he just straight up says "patriotism" here. But really, he's reminding the former colonists that it's more important to be patriotic toward America as a country than their home states. Remember, until the Constitution was written, people's loyalties were generally to their state governments more than a centralized idea of the United States as a nation.
Quote #4
In contemplating the causes which may disturb our Union, it occurs as matter of serious concern that any ground should have been furnished for characterizing parties by geographical discriminations, Northern and Southern, Atlantic and Western; whence designing men may endeavor to excite a belief that there is a real difference of local interests and views. (15.1)
In some pretty spot-on foreshadowing, Washington fears that some people will exploit regional differences to turn those regions against one another. Again, he tries to emphasize patriotism to the whole nation over a local area. It was a good effort, George, but your buddy Abraham (whom you never met) is going to have a lot of regional "difference" on his plate in about 60 years.
Quote #5
In offering to you, my countrymen, these counsels of an old and affectionate friend, I dare not hope they will make the strong and lasting impression I could wish; that they will control the usual current of the passions, or prevent our nation from running the course which has hitherto marked the destiny of nations. (42.1)
Although it's a little depressing that Washington seems to be a bit pessimistic about the future, the idea that he's trying to prevent America's doom is another form of patriotism. He found himself in a position of real power, and instead of taking advantage of it for his own gain, he did what he could to set up the country for success. Even if his efforts only worked a little bit, the intent is just about as patriotic as you can get. That's love for your country right there.