[Is it possible to climb trekking peaks in Tibet in January?]
Yes, for sure!
Winter often offers very clear skies, and the days are not so cold for climbing up to 5500-6000m without other equipment than good shoes and warm clothes. But nights can be very cold if you are camping, so bring a good sleeping bag! High wind can also be a burden, especally in the second half of the winter.
You might find snow in East Tibet, but Central and West Tibet are very dry and alsmost without snow in winter.
If you are interested, I included a few GPS waypoint on a nice trip along the ridge above Karo La (near Yamdrok Tso), which can be done also in winter (max. altitude 5860m)., in the Kaluxung page.