In reply to neuromancer: Head up to Les Diablerettes near Aigle and take the Glacier 3000 cable car up to Glacier de Tsanfleuron. Alternatively, you can drive down to Sion and up to the Col du Sanetsch and access said glacier from there. There are a couple of easy summits in the area. This is probably the nearest sizable glacier with good access.
From Montreux you are still in easy reach of the Swiss side of the Valais (Saas Fee, Zermatt, Arolla) or can get to the top end of the Chamonix Valley fairly easily where the Tour Glacier could be an option for some glacier trekking.