Directly referring to your question, DNA damage does occur as the cold stress first damage the mitochondria which increase the ROS production and decrease the antioxidant activity of the cell. Further, it leaves the cell unable to respond against cold stress or any other stress. That also leads to DNA damage with no energy to repair. Attached articles may help you more.
My dear supervisor, I cant reach up to your level but here is some points that I know or learned in our previous courses .
Answer; Yes in most of cases. Because as we know that whenever there is any stress such as cold stress, ROS are accumulated (free radicals) due to imbalance in metabolic pathways and these ROS denature or ultimately damage biomolecules e.g. proteins, genes, DNA as the result of oxidative stress.
Usually cold stress proceeds as firstly demage the Plasma memebrane, then mitochondria, chloroplast then nucleus genetic material by failure of single transduction that leads to further inactivation of transcriptional factors ( binding proteins) that could be vital for activation of genes in cold stress..
(ROS may toxify and react with transcriptional factors to inactivate and damage the gene or DNA).
ROS may include singlet oxygen, the hydroxyl radical (•OH), one-electron oxidants etc