I know France had some superficial defenses on the border with Italy, but did Italy participate in combat in the Alpine Region/Anywhere else in France?
Italy did...kind of.
I can't give you an interesting story about the battles themselves, but I can tell you a tale of blind incompetence that cost the lives of too many Italians. To understand, Mussolini only declared war against France because he wanted a seat at the negotiating table upon their defeat. In fact, the army was only given two weeks notice because Mussolini needed bodies dead, and *he needed them now.*^1 In his own words on 5 June he is said to have stated, "I only need a few thousand dead so that I can sit at the peace conference as a man who has fought." It is safe to say that Italy only entered into the war for purely political reasons, as the thinking was that if Italy could be perceived as doing something that it would cast doubt onto Hitler to in turn invade them.^2
Predictably little of note happened as the French were dug into defensive positions, and the Italians did as well - under Mussolini's instructions. The Army was ill equipped, undertrained, unprepared, and being led by a man that had assumed supreme command of the military under dubious grounds (Mussolini)^1. Mussolini himself knew how unprepared his military was, as the entire point of the Pact of Steel had been to forestall Italy being dragged into a war until at least 1943^3, and that most military commanders advised that Italy's military capabilities at best suited itself into an offensive with a minor power such as Greece or Albania, but against a larger power should be purely defensive.^2 The boots that the Army went to war with were legendarily bad (the infamous "cardboard boots") and in this 'attack' on France, Italy actually ended up losing more soldiers to frostbite than then French lost in total. (The French suffered 229 total casualties, versus 2,151 lost to frostbite for the Italians^4). It is also to note that Mussolini actively interfered with the invasion as he would go around Army commanders and give detailed instructions directly to their subordinates.^2
As for the fighting itself? It went poorly. Italy declared war on France (and Britain) on 10 June with Army Group West, composed of the First and the Fourth Armies, attacking France on June 21. The day after the French requested an armistice with Italy.^2 By the time of the armistice taking effect four days later (in a move that Foreign Minister Galeazzo Ciano had said was quite fortunate for Italy^5) none of the objectives set out had been met. Mussolini had launched the offensive hoping to secure for Italy Nice, Savoy, Corsica, and French Tunisian ports - and received none of these because Hitler needed the friendship of the now-defeated France instead of Italy^1. Instead Mussolini could only meekly ask for a rough 832 km² occupational zone (almost precisely at where the Italian soldiers were by the time of the armistice) with a 50km wide demilitarized zone - in which Nice fell, outside of Italian control.