Find the syncopation
Jelly, 12 9 bytes

=“e”µ&lt;^\O


As a program, the above code requires quotes around the input. Since that is not allowed, this is a function submission. The output is 1-based. Try it online!

How it works

=“e”µ&lt;^\O    Monadic link. Argument: s (string)

=“e”         Check each character for equality with 'e'. Yields a Boolean array.
    µ        Start a new, monadic chain.
      ^\     Compute the array of partial reductions by XOR, i. e., the parities
             of all prefixes of the Boolean array.
     &lt;       Check if the Booleans are strictly smaller than the parities.
             A truthy outcome indicates an off-beat quarter note.
        O    Yield all indices of 1's.


Update

The above code doesn't work anymore in the latest version of Jelly, since we need a character e, but “e” yields a string. Fixing that saves a byte, for a total of 8 bytes.

=”eµ&lt;^\O


This works as a full program. Try it online!