Program for converting a boring text into a stylish written text
Finally. A program to help me speak the same language as the cool kids. Thanks for that.

Use an array when it's good enough

The second part of convertToTrendyText needlessly uses a StringBuilder instance.
The power of StringBuilder is to efficiently build strings whose size is not known in advance.
In this method we do know in advance,
so there's no need for a StringBuilder,
an array from string.toCharArray() would be more than enough.
After replacing characters in the char[],
you could return it in a new String(...).

Destructive uppercasing

The first part of convertToTrendyText inserts some random smileys.
The second part of the method randomly uppercases some letters.
That risks ruining the following smileys: o.O, x3, xD.
I'm wondering if that's intended or not.
With this potential unintended side effect,
the text might become completely unreadable.

To avoid such destruction,
you could swap the first and second parts:
do the uppercasing first,
and insert smileys after.
That way the smileys will be unaffected by design.

One Random is enough

There's no need to create multiple instances of Random in a program.
It would be better to use just one.
That could be a good step in the direction of making the program testable,
because you will be able to set a seed to get reproducible output.

Getting a random value out of n values

The switch statement in generateRandomSmiley is a bit troublesome.
If you add a new smiley, you have to remember to increment the number in the random.nextInt(...) call, and add a correctly numbered case statement. Such a hassle.

If you use an array of smileys,
then the process of adding or removing values becomes a lot simpler,
more compact,
without having to worry about indexes.

private static final String[] SMILEYS = {
    ":)", ":D", ":*", "&lt;3", "o.O", "x3", ";D", "xD", ":o", ";D"
};

public static String generateRandomSmiley() {
    return SMILEYS[random.nextInt(SMILEYS.length)];
}


Btw, did you notice that there is no case 6 line in your original switch statement?
That leads to getting the same value for it as the default case.
Not sure if that was intentional.
To preserve the behavior of the posted code,
I duplicated the default value ;D at index 6 (in addition to its natural index 9).

A word on performance

Every call like string = string.replace("...", "..."); has to iterate over all the content of the string.
That seems a bit wasteful.
In a toy program like this,
it doesn't really matter,
but it's worth keeping in mind.