what’s the role and responsibility of an internal quality trainer in an entity?

You are right, it's really hard to be defined without background.

traditionally, the content of quality training base on the requirements of qulity plan (Quality Roadmapping),
the responsibility and authority of quality trainer depends the demands of quality function.

For example:

If your company decides to get certification by coming year and your quality function need a trainer to take the basically training for entire employee as well, then this is the main task of this role. if your quality department thinks this is a unnecessary position, you may cancel it accordingly.

according to my experiences, the trainer should have following competences:

• Establish the training plan accordance with the quality roadmapping or quality plan or quality strategy or whatever;
• Build the course ( training meterial, trainee, resource, budget etc.);
• Training
• Follow up the result of training;
• Coaching
• Fast learner, easy cooperation, confidence, good skill of communication and persuading ,independence.

hereinbefore are the general features of a trainer, it's not only suitable for the quality trainer.