The Tsuut'ina is a Northern Athabaskan language, while Apache and Navajo are Southern Athabaskan. The core Athabaskan speaking region is in western Canada and central Alaska, with a secondary center in the American Southwest (the one your seem to be more familiar with) and some smaller outliers on the Pacific Coast (map). The t'ina in Tsuut'ina and the Diné in the Navajo autonym are closely related words, as they both refer to "People." T'ina, Diné, and their related words in other Athabaskan languages are rather iconic of the group, which is why the higher ranks of the linguistic family tree is referred to as "Na-Dené."

The Tsuut'ina are more closely connected with other Northern Athabaskan-speaking people than those in the Southwest though - in particular the Dane-zaa, which has been spelled various ways. They're also known as the Beaver Indians in English, such as on the map I linked. They're #4. The Tsuut'ina are #21, listed by the older English name "Sarcee." One traditional account (now found in the unfortunately named Canadian Savage Folk from the late 1800s) describes their separation from the Dane-zaa:

Only in the traditions of the people can we learn anything of this strange isolation of the Sarcees from their kindred in the far northern country. Tradition says that in the distant past a young Beaver chief shot his arrow through a dog of one of his fellow braves, who was deeply enraged, and vowed vengeance. His friends rallied to his assistance,, and eighty men fell dead as the result of the quarrel. Great was the sorrow in the camp, and a temporary truce was arranged, but sixty people who were friends of the chief who had killed the dog agreed to separate from the tribe and seek a home in another part of the land. They journeyed southward by the shores of the Lesser Slave Lake until they reached the plains and valleys of the Great Saskatchewan.

Another tradition, with more folkloric overtones, was recorded by E.F. Wilson in the mid-1800s in his Report on the Sarcee Indians, describing the Tsuut'ina's separation from the Denesuline (Chipewyan) rather than the Dane-zaa:

Another Indian told us how the Sarcees were at one time one people with the Chipewyans, and gave us the myth which acounts for their separation. "Formerly," he said, "we lived in the north country. We were many thousands in number. We were traveling south. It was winter, and we had to cross a big lake on the ice. There was an elk's horn sticking out of the ice. A squaw went and struck the horn with an axe. The elk raised himself from the ice and shook his head. The people were all frightened and ran away. Those that ran toward the north became the Chipewyans, and we who ran toward the south are the 'Sotenna' or 'Sarcees.'"

(I should mention though, that if you read the full report in the link, there's a 1888 addendum that supports the Dane-zaa / Tsuut'ina connection).

The Tsuut'ina likely entered southern Alberta in the 1700s and by the early 1800s, they had allied with and became members of the Blackfoot Confederacy. The motivation for this alliance was their common enemy: the Iron Confederacy, of which the Plains Cree were a major component. It's debatable whether their relationship with Cree was immediately hostile or something that developed over time due to shifts in the lifestyle (adoption of the horse, for example) which allowed for more intensive utilization of Plains resources and, hence, more competition for those resources. Earlier historians favored the "traditional enemies" angle, while later ones have begun examine the evolving and shifting nature of Plains politics.

As a brief aside, you may be curious about the inverse of question. If the bulk of Athabaskan speaking people are in western Canada, how did the Apache and Navajo come to be in Southwest?

The Apache and Navajo came into the Southwest sometime between 1300 and 1400 CE (at the latest, possibly earlier), having followed the Rockies southward. Mainly this route followed the eastern slopes, along the Great Plains, but a western route through the mountains themselves or into the Great Basin has also been proposed (likely the route was a singular path anyhow). The reason for their long southward trek is not known, but the current reigning hypothesis is that a major volcanic eruption near the southern Alaska-Yukon border (Mount Churchill?) around 800 CE (well remembered in the oral history of the region) triggered a series demographic adjustments that pushed the ancestors of the Apache and Navajo southward.