Hi! So my own history with this area of the country is sketchy (I'm a south-coast historian). But this is somewhat appropriate as the land wasn't so neatly well defined as we would see now on a map as the Alaska Yukon border (though I'm sure I hardly need to tell a flagged AskHist poster that).

Instead, it's best to consider the land in the 19th century as far more fluid. Political boundaries as experienced through spheres of influence were far more sketchy than even than we might initially imagine. Political power was expressed through trading power, and projected from Forts which were mere islands within a vast sea of First Nations power. Conflicts, such as I know of them (and, again - someone correct me if I'm wrong), were individual vingettes of contestation, and never fully escalated into full blown conflict.

Consider, for example, a small local economic 'skirmish' that happened in 1847/1848 in Bella Bella. This is well geographically outside of your geographic area, but was part of that vague northern region where direct political and economic control was highly unstable, to say nothing of the almost non-existent military power.

As recorded by Joseph Mackay, in his Recollections of a Chief Trader, (BCARS - MS-1917, File 32), this HBC fur trader trader mentions a conflict between the Simpseysan [Tsimshian] and the Tongass that he was brought to Bella Bella by the Russian Company to put an end to, having by his authority some sway with the Tsimshian.

However, Mackay takes to opportunity to take advantage of the situation by getting the Russian traders drunk when he hears of a fleet of canoes incoming laden with furs, and proceeds to to more or less screw the Russians out of a large haul of furs through by taking advantage of their inebriated state.

While I suspect you're looking for the usual bang-bang boom-boom conflict, I think the above little story is probably more emblematic of conflict on a highly fluid and liminal border region. Wars, such as they occurred, were probably expressed more through conflicts interceded by First Nations - who ultimately had a lot more control of direct power in the region until later in the 19th century.

While I'm not aware of any Russian-British incidents in the region, in 1804 the Russians came to blows with the Tlingit at Sitka. 12 Russians (and likely many more Tlingit) were killed over several days of fighting. Though this wasn't a major international incident, the British nearly went to war with Spain towards the end of the 18th Century over a conflict at Nootka Sound. The trade in sea otter pelts saw British and American ships trading up and down the coast, along the Alaska panhandle, the BC coast, and all around Vancouver Island. The conflict escalated when Spain asserted sovereignty over the area and impounded British ships trading in Nootka Sound. War between the major powers was a distinct possibility until the French Revolution conveniently put an end to the Bourbon alliance between Spain and France and brought about a negotiated settlement to the Nootka issue.

There was the Ukase of 1821, by which Russia claimed the coast south to 51° N latitude. Britain and the United States protested and the conflict resulted in treaties between Russia and both nations in 1824-1825. I think it is in these treaties that the 54°40' N border comes from. As far as I know the conflict between Britain and Russia was resolved without incident, but Russia did seize an American ship in 1822 for violating the ukase.

Sources...I've read about this in a number of places, mostly books I don't have at hand. Here is a JSTOR article about the ukase (I don't have access to the whole thing though). There's also a bit about it in this book about the Hudson's Bay Company. Also this book and this one.