You should put the single quote inside of the double quotes to string. See the example below
BugFactory bf = (BugFactory)qctd.BugFactory;
TDFilter tf = (TDFilter)bf.Filter;
// List the defect with the status as "Ready for QA"
tf["BG_STATUS"] = "'Ready for QA'";
BugFactory bf = (BugFactory)qctd.BugFactory;
TDFilter tf = (TDFilter)bf.Filter;
// List the defect with the status as "Ready for QA"
tf["BG_STATUS"] = "'Ready for QA'";