GST on supply of labour to institution

Yes, as per language of your query what I could decode is the following

• You have hired a Contractor which supplies labour to you, so this is manpower supply, and thus all payment to that agency by you shall be liable to GST as the relation is Principal to Principal
• Salaries shall be paid then by out of the above amount to the labourers by the agency and that shall not be liable to GST as that is under Employer-Employyeerelationship