There is only one shower of note in the first three months of the year. The Quadrantids have a short sharp peak centred on 04 Jan, and unfortunately this year it coincides with a nearly full Moon so the display will be quite washed out. Expect no more than 20 meteors per hour.
After the quadrantids it is quiet till April’s Lyrids. There are a few minor showers: the γ-Ursae Minorids on 20 Jan (rate <3) is visible from the northern hemisphere, but the α-Centaurids (08 Feb, rate <6) and γ-Normids (14 Mar, rate <6) are southern hemisphere events only unless you’re using a radio detector.
Finally the Anthelion Source is active during this period. This is the name given to a large number of weak, badly defined sources covering a huge patch on the opposite side of the sky to the Sun. You will pick up meteors from this any time between December and September though mostly they will be categorised as Sporadics. Again you’re more likely to pick these up on a radio detector.