How much traffic is there between Montreal and Quebec along route 138, the Chemin du Roy (King's Road)? I know the highway (#40) drains most of the traffic, but I'm wondering if I should take even smaller roads adjacent to 138. I know the shoulder on 138 is safe, but my main concern is really the constant cars passing, not as a safety concern, but just annoyance.
From what i know, i come from that region, the traffic on the 138, while not very dense, is very fast. Peoples drive there as fast as on the highway. I don't where you're going on your trip, but as i plan my tour this summer from Jonquière to the Vermont, i avoid provincial roads as much as i can. If i was you, i'll pass by "La route du Roi" who is the old road who follow the St-Lawrence river. I think the "Route Verte" use that road anyway.
The two times I did the trip by bike, I didn't find the traffic all that bad. It is tough through and out of Montreal, but in places on the 138 it is really light to non-existant. Cars are moving along at a good pace though. It was fine for me. I did the trip two times in the early fall, so the summer peak, might be worse.
What I found late last summer while coming back by car was that several of the bridges that span rivers coming down from the north are missing, leading to long and ugly detours up to the HWY 40 area and back down again. The Bridges were missing from the Quebec end. But after hitting of few of these I went onto the HWY 40 for the rest of the trip. So who know how many in total might have been missing. It looked as though major rebuilding was necesary, so check to ensure that your route is uninterupted. You might think that the few missing bridges were just a fluke, and they may be, but overpasses and bridges have been collapsing in Quebec, which led to an inquiry and an assessment of all such structures, so it is possible to run into multiple affected structures. Indeed I crossed other bridges under construction (and passable) while on the detours.
Fuji Touring 64cm, Peugeot Canada PB-18 61cm and winter beaters
There's only been one overpass collapse in the whole province... "Overpasses and bridges have been collapsing" is a gross misrepresentation.
Anyway, the Québec-Montréal bike route is part of La Route Verte, and as such is clearly marked and so are all temporary bypasses. There's always a non-highway bypass for any type of construction, and there are separate traffic diversion signs for bikes and cars.
La Route Verte doesn't always follow the 138 - it sometimes goes into villages or takes secondary roads that are more pleasant than the 138. So, instead of following the 138 all along, take La Route Verte, which will keep you on the 138 most of the time but also on nicer roads whenever possible.