Happy Fibonacci Day!
J, 58 bytes

3 :''' i_''{~|.|:(,|.)(#&amp;1,2#~0&amp;=)"*}.,0,.(%-.-*:)t.1+i.y'


Uses (%-.-*:)t. for Fibonacci generation. Explanation might come a bit later.

Usage:

   f=.3 :''' i_''{~|.|:(,|.)(#&amp;1,2#~0&amp;=)"*}.,0,.(%-.-*:)t.1+i.y'
   f 5
        ii        
        ii        
      i ii i      
    i i ii i i    
i_i_i_i_ii_i_i_i_i


Try it online here.