What are an employer’s rights if an employee states to their HR department that they are going to

quit but they do not know when?

The employer can treat their notice of resignation/quitting as effective immediately--the employee has stated that he or she does not want to continue to be employed, after all. Or less contentiously, since doing that deprives the employee  of his/her eligibility for unemployment (if you quit/resign, you don't get unemployment benefits), which could prompt a legal challenge by the employee--and even if the employer wins, as they very well may in this circumstances, simply responding to the challenge could be a distraction and cost some money (e.g. for an attorney), the employer could simply terminate the employee immediately: unless the employee had a written employment contract preventing termination in this situation, he or she was an employee at will and could be let go at will. Terminating them means they can get unemployment; while the company may not want that, and would appear to be justified in treating the employee's announcement as an immediately effective resignation, again, a litigious employee could dispute a denial of unemployment or otherwise try to take legal action, and there is a certain value to avoiding potential litigation.