Replace only text inside a div using jquery

Find the text nodes (nodeType==3) and replace the textContent:

$('#one').contents().filter(function() {
    return this.nodeType == 3
}).each(function(){
    this.textContent = this.textContent.replace('Hi I am text','Hi I am replace');
});

Note that as per the docs you can replace the hard-coded 3 in the above with Node.TEXT_NODE which is much clearer what you're doing.

$('#one').contents().filter(function() {
    return this.nodeType == Node.TEXT_NODE;
}).each(function(){
    this.textContent = this.textContent.replace('Hi I am text','Hi I am replace');
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div id="one">
       <div class="first"></div>
       "Hi I am text"
       <div class="second"></div>
       <div class="third"></div>
</div>

Text shouldn't be on its own. Put it into a span element.

Change it to this:

<div id="one">
       <div class="first"></div>
       <span>"Hi I am text"</span>
       <div class="second"></div>
       <div class="third"></div>
</div>

$('#one span').text('Hi I am replace');