It seems to me you and your friend could both be right, depending on how one wishes to use the term “martyr”.
As you know, “martyr” means “witness”, and it’s clear from the Gospel that Christ Himself was a witness. He tells us, for example, that He came to do “not my own will, but the will of the Father who sent me” (John 6:38) and that He does nothing except “what He sees the Father doing” (John 5:19). He thus bears “witness” to the First Person of the Trinity, even as the Third Person bears “witness” to Him (John 15:26).
So sure, Christ is a martyr in that sense, as your friend has proposed.
On the other hand, you’re clearly right too–right in saying that it sounds rather odd, in Christian usage, to apply that word to Christ. The more common and obvious use of “martyr” is in speaking of those who’ve testified (witnessed) to their faith by their deaths. Inasmuch as Christ is God, however, and inasmuch as His death entailed the destruction of death, He’s best construed as the object, not the subject, of martyric witnessing.