Poner una reclamación

Poner una reclamación
 

What s the difference between make a complaint and put in a claim?

..or file a complaint.

To make a complaint is usually vocal.

To file or put in a complaint implies something more formal, perhaps filling out a form or contacting a third party.

By the way, would it make sense to say concertar una reclamacion?

No.
You conciertas una cita.
And what about claim?

file a claim
put in a claim

A claim is a demand for action. It is usually an official form.
A complaint is an expression of discontent, pain, grief or resentment.

When we say that we've filed a complaint, we really mean that we've filed a claim wherein we have stated a complaint and are demanding an action to be taken.

In law, the opening statement (the plaintiff's cause for action) in a civil proceeding is called a complaint.