The Huns, known as the Khün, Hunnu, or Xiongnu in East Asia, migrated from the Altai to the Volga region in the 1st century CE, then invaded eastern and central Europe in the 4th century, establishing the Hunnic Empire (c. 370-469 CE).
There has been an impressive number of studies (Keyser-Tracqui 2003, Keyser-Tracqui 2006, Petkovski 2006, Ricaut 2010, Kim 2010) of ancient DNA from Xiongnu sites in Mongolia, all dating from the Iron Age (300 BCE-200 CE), the exact period when the Huns migrated to Europe. These studies found in total two samples of Y-DNA haplogroup C3 (M217), one N1c1, and one Q, one R1a1a (M17).
45 mtDNA samples were successfully tested and were overwhelmingly East Asian (B4b, C, D4, F1b, G2a), but also had six European haplogroups (13.5%), namely two U2 (including one U2e1), two U5a, and two J1.
In a new study yet to be published by LL. Kang et al. tested three Hunnu samples from Barköl, Xinjiang, China and found that all three belonged to Q1a3a (M3). Interestingly this is the same subclade of Q as the one found among Native Americans.
So far that gives is four Q, two C, one N1c1 and one R1a. The latter two might have come together from the Ural region during the Bronze Age.
While it is very difficult to distinguish Hunnic R1a from other R1a in Europe, C and Q are very easy to isolate. Based on modern frequencies the Huns must have carried far more haplogroup Q than C, although I doubt that it was exclusively Q1a3a. I would rather think that most of the haplogroup Q in eastern, central and northern Europe is of Hunnic origin.