In January and February around Flagstaff, most water will likely be frozen, though it looks like we're forecast to have a rather mild winter. In that area, I'd guess that your best bets would be (in no particular order): Oak Creek near Sedona, the Upper Verde River, and Wet Beaver Creek. I personally haven't fished these waters in a couple years, so I can't tell you how good the fishing should be, but they used to fish okay.
If you're going to be staying in the Flagstaff area, I'd head up to the Ferry; at least you know the water won't be frozen and the fishing should be consistent, if a little slow.
They close the Rim Road on the Rim in the winter, so I'm not sure what will be accessible when you arrive in AZ, and it's probable that most water will be frozen.
Of all the creeks in northern AZ, Silver Creek is probably the easiest to fish and access, as long as a blizzard hasn't rolled through in the past couple days. Just try to avoid fishing there on the weekend; it can get pretty crowded up there then. Also, they stock SC with broodstock for the winter, so if you make it up there, bring some 3x tippet; I've tied into some 7+ pounders in there.
Good luck and good hunting.