"Payroll departament" equivaleria a "departamento ligado a pagamentos/folhas de pagamentos".
employee's department . Está correta essa expressão?
Sim, está correta gramaticalmente. Mas, não me parece uma opção no contexto apresentado.
Personal Department ou Personnel Department.
A melhor opção, a meu ver, é "Personnel Department
>> Personnel Department Course
Recomendo a leitura de um texto retirado de um site americano:
''The Personnel Department
is responsible for recruitment, organizing, selection, and processing hiring of all employees of the Township. The Personnel Department coordinates all employee insurance programs and employee welfare programs including group training such as Supervisory Training, Diversity Training, investment services, and other training. The Personnel Department is also responsible for developing and recommending personnel policies, pay plans, assisting in disciplinary matters, collective bargaining and responding to employee grievances as well as maintaining the switchboard/receptionist services.''twp.west-bloomfield.mi.us