How to set part of text to bold when using AlertDialog.setMessage() in Android?

You need to use Html.fromHtml() too. For example:

AlertDialog.setMessage(Html.fromHtml("Hello "+"<b>"+"World"+"</b>"));

Update:
Looks like Html.fromHtml(String source) has been deprecated in the Latest Android Nougat version. Although deprecation doesn't mean that you need to change your code now, but it's a good practice to remove deprecated code from your app as soon as possible.
The replacement is Html.fromHtml(String source, int flags). You just need to add an additional parameter mentioning a flag.

if (android.os.Build.VERSION.SDK_INT >= android.os.Build.VERSION_CODES.N) {
   AlertDialog.setMessage(Html.fromHtml("Hello "+"<b>"+"World"+"</b>", Html.FROM_HTML_MODE_LEGACY));
} else {
   @Suppress("DEPRECATION")
   AlertDialog.setMessage(Html.fromHtml("Hello "+"<b>"+"World"+"</b>"));
}

For more details have a look at this answer.

This page describes how to add HTML formatting to resource strings.

<resources>
    <string name="welcome_messages">Hello, %1$s! You have &lt;b>%2$d new messages&lt;/b>.  
    </string>
</resources>

And do not forget to use: Html.fromHtml

AlertDialog.setMessage(Html.fromHtml(getString(R.string.welcome_messages)));

This works for me