I was wondering if during the age of colonization, primarily from Europe to the Americas, there was ever a group from one country setting out to colonize a stretch of land, only to run into a fledgling colony by one of their rival European powers?
You might look into the conflict over the Red River settlement in British North America. Briefly, in 1811, Lord Selkirk, a major shareholder in the Hudson's Bay Company, decided to help a bunch of crofters that had been forced off their land in Scotland by enclosure. He moved them to land that was already claimed by the Northwest Company, a fur trading company based in Montreal. This led to open warfare, and Selkirk even brough in Swiss mercenaries to protect the settlement.
Although this isn't exactly what you asked, you may want to look into the Darien scheme. It was the only attempt by the Kingdom of Scotland to create a colonial empire. They tried it in the Isthmus of Panama, which happened to be right in the middle of Spanish New Grenada. There were a fair few battles fought, and the Spanish were one of the many reasons the Scottish colony failed.
I would try to go into more depth, but my knowledge on this only comes from my high school history classes, and I'm pretty sure they probably aren't going to cut it here.