Cache DOM selectors

You select the target container over and over again. You should get it once outside the function, cache it in a variable and reuse it:

const target = document.getElementById('quotes');

Naming

Try to use descriptive names instead of myarray, myVar and quote:

const quotes = [];
function changeQuoteRandomly() {}

Style

It's easier to read, if you use single quotes on the strings, as you don't have to escape the double quotes:

quotes.push('"Important words" by Someone');

You sometimes use a semicolon at the end of the line and sometimes you don't – use only one way and be consistent:

var myarray = []
var myVar = setInterval(quote, 15000);

I would prefer using push to add elements to the array instead of setting the index manually. Or you could even add all elements during initialization:

const quotes = [
    '"Important words" by Someone',
    '"Other important words" by Someone'
];

Both ways make it easier to re-arrange elements later as you don't have to keep track of the index yourself.

Semantics

Currently your markup looks like this:

<div id="quotes">
    "The single best piece of advice: Constantly think about how you could be doing things better and questioning yourself."<br>
    <strong> Elon Musk</strong>
</div>

You could improve this, by using blockquote, cite and even figure elements. Here you can find more infos and examples on W3C: 4.4.5. The blockquote element.

User experience

While this is a nice addition on your website, there are a few downsides:

• There's no introduction to this section, the visitor has no clue why you're showing them. Try to relate them to your business.
• The first quote is shown after 15s, until then only an empty blue container ist visible.
• 15s between quotes is a really long time. There's no indicator that more is coming, so there's a great chance that a user will not see a second one.