It can snow in the mountains any month of the year. I've skied two feet of powder on Memorial Day, have seen light snowfall in August and have trudged through high-elevation snow in September.
Which is a nice way of saying that by October, you probably want to pick your route carefully. There is no route from Northern NM through much of Colorado into Utah that does not involve some very high mountain passes.
You're going to need to check the weather every day and be prepared to adjust accordingly and head to lower ground. You might be forced to stay far south and west, cutting across through the four corners region, without getting far into Colorado. It's not a horrible place to be "forced" to ride through. Once you're on the Utah side of the mountains, things get a lot easier. I actually like Southern Utah that time of year. Rain is definitely possible, but things are cooler than the summer months and the tourists are mostly gone.
But remember that even in the Rockies, and even in the heart of the winter, most days are sunny days. In October, so long as your schedule and route are both a bit flexible and you exercise appropriate levels of caution and planning, you should be fine.
I would bring some colder-weather gear though. Pretty much any route you take through Colorado will have you above 10,000' at some point and by October those segments can get fairly chilly even in the absence of snow.