Within Fibonacci Numbers

Jelly, 10 bytes

0ÆḞ©w¥1#;®

Try it online!

How it works

0ÆḞ©w¥1#;®  Main link. Argument: n

0           Set the return value to 0.
       #    Call the second link to the left with arguments k = 0, 1, 2, ... until
      1     one match has been found.
     ¥        Combine the two links to the left into a dyadich chain.
 ÆḞ             Compute the k-th Fibonacci number...
   ©              and copy it to the register.
    w           Yield 1 if n occurs inside the Fibonacci number, 0 otherwise.
         ®  Yield the value stored in the register.
        ;   Concatenate the index and the Fibonacci number.

Python 2, 56 bytes

f=lambda n,i=0,a=0,b=1:`n`in`a`and(i,a)or f(n,i+1,b,a+b)

Try it online!

Perl 6, 30 bytes

{first :kv,/$_/,(0,1,*+*...*)}

Try it online!

first is a function that returns the first element of a sequence that passes a test, and it conveniently takes a :kv adverb that tells it to return both the key (index) and the matching value.