Secure MAC implies that probability of same tags on different messages is negligible
Is this a homework problem?

Here's a hint: Suppose that the probability of getting the same tag on two different messages is $p$.  Show how to construct an adversary that breaks the MAC (i.e., forges a valid tag), and that has success probability $p$.  Assuming the MAC is secure, what can you conclude about the possible range of values for $p$?