Each journey in this satirical work by Swift represents a specific "conflict". There are four journeys that Gulliver undertakes and each new society that he encounters causes him to take a look at his own English society. These insights into the politics and human nature create quite a controversy. He uses each new society as a contrast to his own and also demonstrates how each group cannot believe there is a different way of living or any other way of life that could be better.
You can review the enotes review of Gulliver's Travels for a more expansive review of each journey and decide which conflict you wish to focus on for your report.
It would be difficult to state there is just one conflict unless you want to focus boradly on the idea of how Swift saw society as narrow minded and even the ugliness of human nature.
This work is a satire. Swift uses the personna of Lemuel Gulliver, a ship's doctor (for credence), to satirize so many issues in the English government. Several conflicts which show throughout the four-book work are the way the government officials handle things like elections, wars, and the affairs of the country--both domestic and foreign. He also makes fun of the scientists of his age, illogical behavior, and the vanity of humanity. There are many conflicts, but perhaps you could narrow it down to one major conflict in each book.
Check out the enotes summaries and choose the conflict you want to tackle for your book report in each section of this piece of literature. Good Luck! Sounds like a fun project.