Assuming one indeed talks about quaternionic representations, it was answered in Which groups have only real and quaternionic irreducible representations?

And indeed $SU(2)$ is the only example. The notation used there denotes $SU(n+1)$ by $A_n$.

Assuming one indeed talks about quaternionic representations, it was discussed under a more restrictive assumptions (all irred. characters are real) in Which groups have only real and quaternionic irreducible representations?

And indeed $SU(2)$ is the only example in their case. The notation used there denotes $SU(n+1)$ by $A_n$.

I gather the answer to the question here must be known, but it needs some reading to be done along the lines discussed in the link.