Based on the assumption you are already able to fetch the data (quickSearchResults), a parsing function would look like the following:
function findElementInData(dataPoints, nameOrSymbol) {
    return dataPoints.filter(point => point.name == nameOrSymbol || point.symbol === nameOrSymbol)
                     .map(point => {
                         return { name: point.name, rank: point.rank }
                      });
}
var elementsByName = findElementInData(quickSearchResults, 'Bitcoin');
console.log(elementsByName); // [{name: "Bitcoin", rank: 1} ]
var elementsBySymbol = findElementInData(quickSearchResults, 'BTC');
console.log(elementsBySymbol); // [{name: "Bitcoin", rank: 1} ] 
This will give you all elements matching either the name or the symbol passed for the given array of data.
Update: It appears you do not have access to the data yet.  This is a full example:
function findElementInData(dataPoints, nameOrSymbol) {
    return dataPoints.filter(point => point.name === nameOrSymbol || point.symbol === nameOrSymbol)
                .map(point => {
                    return { name: point.name, rank: point.rank }
                });
    }
function getData(dataUrl) {
    return fetch(dataUrl)
    .then(res => res.json())
}
window.addEventListener('load', () => {
    getData('https://s2.coinmarketcap.com/generated/search/quick_search.json')
    .then(dataPoints => {
        console.log(dataPoints);
        let myResultsBTC = findElementInData(dataPoints, 'BTC');
        let myResultsBitcoin = findElementInData(dataPoints, 'Bitcoin');
        let myResultsEtherum = findElementInData(dataPoints, 'Ethereum');
        let myResultsETH = findElementInData(dataPoints, 'ETH');
        console.log(myResultsBTC);       // {name: "Bitcoin", rank: 1}
        console.log(myResultsBitcoin);  // {name: "Bitcoin", rank: 1}
        console.log(myResultsEtherum);  // {name: "Ethereum", rank: 2}
        console.log(myResultsETH);      // {name: "Ethereum", rank: 2}
    })
    .catch(err => {
        console.error(err);
    });
});