Solution 1:

I think the following should work:

// fire request
request({
    url: url,
    method: "POST",
    json: requestData
}, ...

In this case, the Content-type: application/json header is automatically added.

Solution 2:

I worked on this for too long. The answer that helped me was at: send Content-Type: application/json post with node.js

Which uses the following format:

request({
    url: url,
    method: "POST",
    headers: {
        "content-type": "application/json",
        },
    json: requestData
//  body: JSON.stringify(requestData)
    }, function (error, resp, body) { ...

Solution 3:

You don't want multipart, but a "plain" POST request (with Content-Type: application/json) instead. Here is all you need:

var request = require('request');

var requestData = {
  request: {
    slice: [
      {
        origin: "ZRH",
        destination: "DUS",
        date: "2014-12-02"
      }
    ],
    passengers: {
      adultCount: 1,
      infantInLapCount: 0,
      infantInSeatCount: 0,
      childCount: 0,
      seniorCount: 0
    },
    solutions: 2,
    refundable: false
  }
};

request('https://www.googleapis.com/qpxExpress/v1/trips/search?key=myApiKey',
        { json: true, body: requestData },
        function(err, res, body) {
  // `body` is a js object if request was successful
});