First let me refactor your two procedures that have a mix of promises and async/await so that they have just equivalent async/await.

const getNumberOfQuestions = async () => {

    const message = await this.channel.send('How many questions should I ask? (1-10)');

    try {

        const collected = await this.channel.awaitMessages(message => message.author.id === this.owner && !isNaN(parseInt(message.content)),  { max: 1, time: 15000 });

        this.channel.send(`You asked for ${collected.first().content} questions.`);

        return parseInt(collected.first().content);

    } catch(collected) {

        this.channel.send('You did not tell me how many questions you wanted. Ending the quiz.');

    }

};



const getDifficulty = async () => {

    const message = await this.channel.send('What difficulty would you like: easy, medium, hard?');

    try {

        const collected = await this.channel.awaitMessages(message => message.author.id === this.owner && ['easy', 'medium', 'hard'].includes(message.content.toLocaleLowerCase()),  { max: 1, time: 15000 });

        this.channel.send(`You asked for ${collected.first().content} difficulty.`);

        return collected.first().content;

    } catch(collected) {

        this.channel.send('You did not tell which difficulty you wanted. Ending the quiz.');

    }

};

As you can see, awaiting on a promise is like treating what's after it as inside a then, where the resolved value is what's returned by the awaited expression. The rejection (.catch()) of the promise is translated to an exception which can be try{...}catch{...}ed.

Knowing this, what you're doing by calling

getNumberOfQuestions();

getDifficulty();

is calling both functions asynchronously - without waiting for the promises returned to settle. What you may want to do instead is to await the first's end before invoking the second. Like so:

await getNumberOfQuestions();

await getDifficulty();

However, the await keyword makes sense only in an async function, so you can do it like so:

(async()=>{

    await getNumberOfQuestions();

    await getDifficulty();

})();

More about async functions: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Statements/async_function