Trekking is possible any time of the year depending on where you are going. The popular seasons are spring and autumn. During winter, trekking is possible at lower altitudes. During the monsoon season, you can trek in the rain shadow areas north of the Himalaya like Mustang, Upper Mustang, and Dolpo. These places are out of reach of the rain clouds because of the high mountains and are unaffected by the monsoon.
December and January are the coldest months with heavy snowfall in the upper hills, crossing high passes at this time can be impossible. February and May are warmer in lower altitude but in high hill can still be very cold and heavy snowfall. In raining season June and July, is just possible for short trekking because of dangerous flooding. From September to November is the most suitable season for trekking as rain is at a minimum, Sky seems clear.