The hash (#) for date searches is only support in find/search, not in table filter.
(Noet thar the hash as case-insentive search is supported in filters).
You can use the between operators to find all messages of the day.
The object arguments needs to be an array of 2 date objects:
var curDate = new Date(); // Get the new date object
curDate.setMonth(0,1); // Set the date to 1st Jan of the year
foundset.addFoundSetFilterParam('date_column_name', '>=', curDate);
foundset.loadAllRecords();
Good idea, but be sure to set it really to 1st January at 00:00:00, setMonth() sets only the month, not the time.
So you would miss the (small) time range up to the current time on 1st January.