"Mein Haar braucht einen Schnitt?", would be understood, but sounds a bit funny. Kind of like a word play that a native would use. Normally one would say something like: "Meine Haare müssen geschnitten werden." or "Ich möchte meine Haare schneiden lassen." However if you enter a Friseursalon, this is usually assumed. The main information to transmit is what exactly you want. First of all you would want to pick among: Schneiden, Waschen, Färben...., which you can combine with und or nur. For instance you could say:
Nur Schneiden bitte.
Waschen und Schneiden bitte
Assuming that with "I just need a trim, please.", you mean that they only need to do a small "refresh", that would be: