Yes they do. More precisely, Guoguang bus 1819 (Kuo-kuang, "King Bus", white and purple livery) is a 24h service. The departure times from T2 are (add 10 minutes for T1):

00 :00 :15 :25 :35 :50 :55(2)
01 :00(1) :05 :15(2) :20 :35 :50
02 :05 :20 :50
03 :20 :50
04 :20(1) :35(2) :50

There is some discrepancy between the two sources, which I marked with (1) or (2) respectively:

1. Guoguang
2. Taiwan Bus (official schedule aggregator for all bus operators)