If you are referring to a company providing a calibration service to testing laboratories (i.e. they are a calibration laboratory) then yes, they need to implement and be accredited to ISO 17025.
If you are asking if it is compulsory for a company providing a testing service (i.e. they are a testing laboratory) and they only calibrate the equipment they use for those tests inhouse, (i.e. they do not use an accredited calibration service), then yes, they need to implement and be accredited to ISO 17025 for the calibration activity to ensure metrological traceability of their test measurements. The testing activities themselves will need to be accredited if the sector, customers or legislation require it. For many sectors, however, it is voluntary to be accredited for the testing activities.
If the company is not a calibration or testing laboratory but using equipment for example, for manufacturing, and they are calibrating their equipment themselves (in-house), the type of sector will dictate what accreditation is required. For example, for any pharmaceutical production, Good Manufacturing Practice (GMP) accreditation will be required for the production activities.
In any case, the confidence in the quality of the results is achieved through implementing a management system based on ISO 17025 for calibration activities. The traceability of the measurement must be ensured through a formal unbroken chain of calibrations using references traceable to international measurement standards.