I called an agency to get the name of the head of HR or the person to whom I should address my cover letter. But the receptionist would not tell me or transfer me to the department. She instead said I should address the letter to "Hiring Manager". I never feel good doing this. It seems lazy and unprofessional. But is it really okay?

Does the agency have a website? Usually, on the agency's website there would be a "staff" page that lists all of the agency's staff or at least its managerial staff. From there, you'll probably be able to see to whom you should address your cover letter. In most nonprofit organizations there's only one hiring official, and that's the executive director.

It really it OK to address letter to Hiring Manager if you can't get the name; we generally don't give out that information but if someone is resourceful and checks out the website, they will find out who the right person is.

Many companies does not post their executive staff profiles on the website. I would just go with Dear Sir/Madam:

It is an outdated practice to address a letter Sir/madam. Hiring manager is what most people are doing now.

Any time an organization tells you to do something specific with regard to their hiring process, it's okay to follow their instructions. Each organization has its own process.