I know this is an old question, but maybe I can be of help to someone, bc this also stumped me until I read the documentation a bit more:
prevArrow string (html|jQuery selector) | object (DOM node|jQuery object) Previous Allows you to select a node or customize the HTML for the "Previous" arrow.
nextArrow string (html|jQuery selector) | object (DOM node|jQuery object) Next Allows you to select a node or customize the HTML for the "Next" arrow.
this is how i changed my buttons.. worked perfectly.
$('.carousel-content').slick({
prevArrow:"<img class='a-left control-c prev slick-prev' src='../images/shoe_story/arrow-left.png'>",
nextArrow:"<img class='a-right control-c next slick-next' src='../images/shoe_story/arrow-right.png'>"
});