Class for formatting topics using BBCodes
First of all you make 1 big fault by saying that your classes inside MarkUp are private.
They aren't, they are package private, more info on site of oracle.

Then, let's start with the enum.
Your final String code is also package private. => change them to private.
Your getter for the code is also a little off the normalisation, let's try to call it getCode.

Final result could be :

private enum BBcode {

    BOLD("B"),
    ITALIC("I"),
    IMAGE("IMG"),
    UNDERLINE("U"),
    VIDEO("VIDEO");
    /* many others... */

    private final String code;

    private BBcode(String code) {
        this.code = code;
    }

    public String getCode() {
        return this.code;
    }
}


The class Syntax you could add constants in stead of repeating your chars.
Then also, you have repeating stuff what you can put together.
The naming in, out are for me to broad. 

Example of how I should make the class :

private class Syntax {

    private static final char START_OPENING_TAG = '[';
    private static final String START_CLOSING_TAG = "[/";
    private static final char CLOSING_TAG = ']';
    private static final char EQUAL_SIGN = '=';

    public String getOpeningTag(BBcode bbcode) {
        return addStartOpeningTag( bbcode.code()); // will return "[BOLD]" for example.
    }

    public String getOpeningTag(BBcode bbcode, String value) {
        return addStartOpeningTag(bbcode.code() + EQUAL_SIGN + value); // will return "[COLOR=#333]" for example.
    }

    public String getClosingTag(BBcode bbcode) {
        return addStartClosingTag( bbcode.code()); // will return "[/BOLD]" for example.
    }

    private String addStartClosingTag(String value) {
        return START_CLOSING_TAG + value + CLOSING_TAG;
    }

    private String addStartOpeningTag(String value) {
        return START_OPENING_TAG + value + CLOSING_TAG;
    }
}


As you can see now, I can easily refactor the code.
Let's say you class grows and we need to change the ] to &gt;
For me it's one little change and it's completly done.
Your old class, you have already to change it 3 places, possible fault's .(forgetting one to change) waiting to happen when your class grows.