To include quotes, just include them in the input string. Like this:
C#:
imap.Search(ImapSearchParameter.Subject("ID=\"1122\""))
VB.NET:
imap.Search(ImapSearchParameter.Subject("ID=""1122"""))
However, please note that it depends on the IMAP server how it will interpret the parameter.
For example Gmail searches only for whole words in subject. E.g. Gmail matches "VENDOR" but not "VEND".