I think that this is a bit inefficient, since it will continue to run through each unquoted character even indexOf found the start.

Also I am concerned that the following string would find a false positive:

abc '{{' }} def

since the {{ is in a string, it should be ignored but would be picked up by the indexOf(), when the charIndex is 0.

I think this would also be clearer if you created a new function _getInterpolationStartIndex(input, expressionStart, start):

  private _getInterpolationStartIndex(input: string, expressionStart: string, start: number): number {
    let result = -1;
    this._forEachUnquotedChar(input, start, charIndex => {
      if (input.startsWith(expressionStart, charIndex)) {
        result = charIndex;
        return true;
      } else {
        return false;
      }
    });
   return result;

Then you could do:

const startIndex = this._getInterpolationStartIndex(input, start, 0);
if (startIndex === -1) {
  return;
}
const endIndex = this._getInterpolationEndIndex(input, end, startIndex+start.length);
if (endIndex === -1) {
  return;
}

this._reportError(...);

I didn't want to add a _getInterpolationStartIndex, because there's some other logic that looks for an interpolation start in splitInterpolation, but the requirements there are slightly different. As for the abc '{{' }} def case, I think I tried something similar and it was reported as a different (syntax?) error further down the compilation process.