Do you put quotes in italics in an essay

Simply put: no.

APA's Publication Manual (2020) indicates that, in the body of your paper, you should use italics for the titles of:

• "books, reports, webpages, and other stand-alone works" (p. 170)
• periodicals (journals, magazines, newspapers)

Beyond APA's specific examples, know that certain types of titles are almost always written in italics. 

A general rule of thumb is that within the text of a paper, italicize the title of complete works but put quotation marks around titles of parts within a complete work. 

The table below isn't comprehensive, but it's a good starting point

On an APA-style reference page, the rules for titles are a little different.  In short, a title you would italicize within the body of a paper will also be italicized on a reference page.  However, a title you'd place in quotation marks within the body of the paper (such as the title of an article within a journal) will be written without italics and quotation marks on the references page.

Here are some examples:

Smith's (2001) research is fully described in the Journal of Higher Education.

Smith's (2001) article "College Admissions See Increase" was published in the Journal of Higher Education after his pivotal study on the admissions process.

Visit the APA Style's "Use of Italics" page to learn more!

Normal style in narrative writing is to use quotes, not italics.

Some authors of fiction, notably James Joyce and William Gaddis, used an em dash at the beginning of a paragraph that started with a quote:

— Do you have the money? he asked.

But that was an idiosyncratic usage. Since this is an element of style, you are free to do whatever you wish, but remember that you may only wind up confusing your audience if you stray too far from what they perceive to be customary.

Some authors use italics to denote what a character may be thinking:

"Do you have the money?" he asked. He didn't have it. She was sure of it.

Edit: Jen is asking about how to quote something a character is writing. Here is how I have mainly seen it done, with the entire written text offset. Note that I'm not good enough at formatting in these answers to make it exact, so I'm putting everything into a blockquote and then formatting the written part as code. But on a real page the background would be all white:

Jen took a pen and began writing:

This is the first time I have written to you. I would like to ask you
if you might be interested in a manuscript that has come into my
possession, one which

She put down the pen. "Dammit," she muttered under her breath. What could she say to pique Roderigo's interest?

(The last paragraph would not be indented, but I can't seem to make that work with the formatting tools available to me here.)

The point is, representing writing on a written page is a different convention from representing speech or thought. Sometimes, if the writing consists of only a few words, a different font or use of uppercase lettering does the job:

Mary yelled, "I know what we need. A sign we can hang on the front of our clubhouse!" And she picked up a marker and a piece of cardboard and wrote: NO BOYS ALLOWED!!! She put down the marker and surveyed her work, then displayed it to the other girls. "What do you think?"

Do you italicize a quote in an essay?

Should my quotes be in italics?

What words do you italicize in an essay?